Hello I agree with Ria's points
We were all new hamster owners once and found out the hard way!
I do hope she recovers soon. It is unusual for a Syrian to bite unless they are in pain or very frightened. Or - another reason could be - if you smell of food! It sounds obvious but always wash your hands before handling them and don't use honey or flower scented soap! Just plain soap. If you've just eaten a bag of crisps your finger may taste like food
I hope the vet finds the cause. It is possible she was injured if landed awkwardly but seems to be recovering. In terms of handling, best to do so "low" - over something soft - so over the sofa or on the floor lower down so she can't fall far.
I also agree that she may not have had enough nutrition - the amount seems small, the mix is rubbish (sorry! not criticizing you but criticizing the mix). And if you are cleaning her out every week and removing any hoard that will stress her a lot and remove her source of food.
Again not intended as criticism but many owners get told all sorts by pet shops.
The key is to have their environment right so they can have normal behaviours - then there is a lot less cleaning needed and they can feel secure having hoards to snack from. They rarely actually eat food from their bowl - they tend to pouch the food from the bowl, hoard it, and snack from their hoards. So if hoards were being thrown away with cleaning she may actually not have been getting much food at all. Also when food is short they sometimes eat their poops (which is normal - they have two stomachs and can redigest vitamins etc from their poops) - so if all the poops were thrown away as well .....
Stress is something that can lead to illness with them as well and they do find cage cleans extremely stressful or any major change to their environment.
My tips are:
1) Switch to a different mix (which you've already said you are going to do). But it needs to be one specifically for Hamsters - not one for various species like hamsters or gerbils. As a basic mix you can't go wrong with Tiny Friends Harry Hamster - it contains everything needed, has the right amount of protein and is sugar free (they can get tooth decay as well!). It's not that expensive either.
Over a week or two. Add a bit of the new to the old and you can be switched over within two weeks. By the end of week one it would be 50/50. About a level tablespoon a day. She may not take it all so you can then top it up the next day. I find they won't eat "old" food though, so after two days, chuck it and put another tablespoon in - it seems a waste but it's not that expensive.
They also need a bit of fresh veg every day. Just a really tiny amount - say 1cm cubed size (doesn't have to be cube shaped). Cucumber and broccoli are favourites and both safe. Also a pea or two (if you've cooked peas for your dinner), a bit of carrot etc. Avoid cabbage and lettuce - they are watery and can lead to diarrhoea.
A bit of variety by giving a different one each day or 3. I find cucumber and broccoli keeps longest in the fridge. Start by giving it every third day until her digestion has adapted. Then every day after 10 days to 2 weeks. I can see you're already giving her veg
Quit the Timothy Hay - too much roughage for hamsters and can upset their digestion and absorption of food.
2) Ideally switch bedding. Kaytee Clean and Cosy (non scented) is good. I use Fitch which is very similar but works out a bit cheaper (comes in bulk in 10kg bags) - you need more for a larger cage but actually use less because you don't need to clean out nearly so often. Fitch is recycled food grade paper.
3) Litter tray - better to use Chinchilla Bathing Sand - it soaks up the pee and is safe - potty litter, if ingested can make them quite ill and cause digestive blockages.
4) Nesting material - as mentioned above - paper is the safest - just use plain white toilet paper. Tear a few sheets into strips and have a pile of it in the cage but not inside the house - it's all you need. As it goes down, top it up now and then. The hamster will pouch it and take it to build a nest. Their raison d'etre is nesting, hoarding and foraging - foraging for bedding and food. So scattering the odd bit of food as well as putting it in the bowl is good so she enjoys finding and foraging for it. No fluffy bedding or cotton wool type bedding - these have led to serious digestive blockages and death.
5. Cage - for a Syrian the best type of cage is one continuous floor area rather than two cages joined together. The cage is their "habitat" and the house/nesting box is their nesting place. In their habitat they need normal behaviours and areas for different things. And enough space (especially at night when they're most active). About the minimum size floor area for a single cage should be 80cm by 50cm - something like the Hamster Heaven cage. Many people prefer 100cm by 50cm because Syrians need larger toys, tunnels and houses. Pets at Home stock the Savic Hamster Plaza which is a 100cm cage with a large front opening door.
Personally I prefer cages to tank styles - the ventilation is better, you can attach things to the bars and it helps with interaction with the hamster. Also front opening is often better if possible, rather than top opening. The paw hut is a good size but some hamsters would chew through it - and it doesn't help with interaction and taming.
I would go for the Hamster Heaven or Hamster Sky (as an absolute minimum) or the Savic Plaza. The HH needs most of the plastic stuff chucking! Tubes and houses not big enough for a syrian and the tubes and penthouse encourage unnatural behaviours (like blocking off tubes and peeing in the house.
6) Environment and behaviours. The best basics are - a good sized floor area in one cage that is not too tall (they fall easily and are ground dwellers and diggers). Minimum 4 to 6" depth of substrate. A house or nesting box that is a good size - large enough to build a large cosy nest. This is where the savic plaza is good - plenty of space to put a large house in it. They need a house that is dark inside as well. So a bendy tunnel over the door helps with that. If it's big enough you can also put the litter tray inside - they instinctively like having an ensuite toilet and use it! If you put it in the right place! (Usually the door end at the back - they tend to nest at the other end as it's darker). A shelf or platform - something to sit under and climb onto - it's also a good place to put heavier items like ceramic items - otherwise they can be tunneled under and squash them. Floor toys like tubes and tunnels (pringles tubes are a good size - cleaned out of course). And hidey places - little hideout dens etc. Cardboard tissue boxes are fine - doesn't have to be expensive. Coconut hideouts are popular as round. Some overhead cover as they don't like to feel exposed above (being prey animals). A shelf provides that partly but hanging sputniks are good too. If placed next to a shelf they are easy to use and provide overhead cover as well. Rat sized tunnels and sputniks are best for syrians.
An upright wheel - 28cm or more ideally.
Then you have a happy hamster with an enriched environment. Enrichment is the key - variety - places to go, things to do, different textures, favourite spots.
As she has had wet tail before as well I would consider giving pro biotics for a while - but check that with your vet. I use benebac for small animals but Dr S from rat rations is also recommended by some owners.
This helps their digestive system become balanced and a hamsters digestive system can be affected a lot after antibiotics - affecting their ability to absorb nutrients properly.
Cleaning. With a good sized cage and 4 to 6" substrate and a litter tray, there is no need for weekly cleans. You can "spot clean" mostly - ie take out the odd handful of soiled substrate and replace it with another handful. But if she uses a litter tray then the only real need is to empty that once or twice a week - and occasionally pick out poops if they start taking over the cage but otherwise they are not dirty or smelly - they are like little seeds (and as mentioned they hoard or eat some sometimes).
I go about 3 months before doing a substrate change and even then only replace about two thirds of it - in an 80cm cage. So I keep the clean third (usually the bottom bit) and spread that on top of the new substrate. That way it smells familiar and doesn't stress them.
In between you can clean any other items as and when - eg wipe the wheel one week, clean any toys etc another week. Toys don't need doing that often.
They are actually very clean little things - it is only their pee that is smelly/soils things.
There is no need to use pet disinfectant unless there has been illness or disease (which at present their has) - and always best to rinse well afterwards if using it to minimise any scent. A hamsters scent is many more times powerful than ours and strong scents can affect their sensitive respiratory tracts. So no scented anything (granules, bedding etc).
When no disease or illness just warm soapy water is fine for the cage base (bit of fairy liquid) and I often just use a warm damp cloth because the cage base is clean and dry under the substrate.