I agree with changing her over the sooner the better. Put lots of substrate in the base of the new cage - at least 3 to 4" deep but the deeper the better! I fill our cage base to the brim (that encourages them to dig their own tunnels which is supposed to be a big stress reliever). Then as Shannon says, let her play in it for a while for say 2 or 3 evenings. Then on the fourth evening, pop her somewhere safe (eg a pet carrier) and move all the bedding from the old cage on top of the bedding in the new cage and mix it all up together (so it smells familiar). Then the wheel and toys as Shannon says. It's important not to clean anything before moving the stuff, and not for two weeks after the move. Any stinky bits will get mixed in with the new bedding and be fine for a couple of weeks. I would recommend getting a hamster potty litter tray and when you know which corner is her wee corner in the new cage, put the litter tray in that corner and she'll probably just use it. It means the rest of the cage stays mostly clean and you don't have to do full cleans so often, and just empty out the potty tray once a week, wash it out and refill it. This worked so well for me I could go 3 or 4 months without needing to do a full cage cleaning. Cleaning is one thing that totally stresses hamsters out as it removes all their familiar scents and trails they have made in their cage and instinctively they think a predator or another animal has been in their cage and get very anxious and even aggressive.
I think the bigger cage and a good depth of substrate will make a huge difference. They also need a 27 to 30cm diameter wheel. If they run in a wheel that is too small their backs are bent and this can cause pain (which also makes them stressed and aggressive sometimes and can lead to vet bills for spine problems).
So you are on the right track. Alaska cage, good sized wheel, plenty of substrate, some kind of house/hide and a few toys. Incidentally the cats eye house that comes on the shelf in the Alaska cage might need taking out as the holes are probably too small for a syrian (and they can get stuck in things!). The shelf itself is ok.
This wheel would be fine and fit well in the Alaska. You'd need the 28cm one and if you order it at the same time as the Alaska you'll get free postage on them both.
Small Pet Exercise Wheel | Free P&P on orders £29+ at zooplus!
If you want a good toy/house to put in, ours loves this cardboard house that has hay stuck on the outside and carrot on top. He scratches away inside it pulling off the hay and sits on top nibbling the carrot. Keeps him happy for ages. It might even do as a main house for nest building. I put mine in the freezer for 2 days before putting it in the cage, as I worry about hay having mites. The freezer for 2 days will kill anything!
Rosewood Naturals Carrot Cottage Guinea Pig House, Medium: Amazon.co.uk: Pet Supplies
Then you just need lots of strips of torn up sheets of white toilet paper in a big mound somewhere in the cage, so your hamster can pouch it and build a nest somewhere.
I am sure her stress levels can be reduced a lot. Their number one most important thing is their nest. They don't like it being disturbed. If it isn't wee'd in I leave it and don't touch it. If it is stinky and wee'd in you might have to remove some of it when doing a cage clean, but always leave some of the original nest behind and put more white toilet paper out so she can rebuild it. The number two most important thing is their wheel - they love their wheels, and if she doesn't have a 28cm wheel at the moment, she will go crazy for the one linked as it will be easier to run in and much more comfortable for her.
And space - so they can roam, find places to hide and wash and bury their hoards, and just explore.
Toys wise, I would avoid any hanging toys (except fixed things like the sputniks) or a grass hammock is ok, but I think anything with chains or rings is a hazard and hams can end up getting a foot caught jumping off and end up hanging by a broken leg.
This grass hammock is quite good as a roof toy, maybe near the shelf so she can get into it easily, or near the house roof. Again if you order it from zooplus at the same time as the cage you won't pay postage on it.
Natural Grass Hammock for Pets at zooplus
With a hammock, a wheel, a house, the shelf and plenty of substrate, she would be fairly sorted, although adding a tube to the floor of the cage would be a good toy. A kitchen roll tube is fine if she isn't too big, or one of the medium sized chew tubes
Boredom Breakers Chew Tube Medium Pets Small Animals: Amazon.co.uk: Electronics
These bendy stick bridges are also good for hiding under and climbing over, or to make a tunnel to a house entrance. I put those in the freezer for two days as well lol! Just in case of woodworm. But I don't think many people do that. I'm just cautious!
Wicker Bridge for Small Pets at zooplus
On the day you move her over, try and have everything set up ready - toys, wheel, house and everything in place, so you don't have to disturb her by making changes too much for the first couple of weeks, then leave her alone for 2 or 3 days. It can take about 10 days for them to feel at home in a new cage. Ours got really nervy and shaky when he moved from a smaller cage to a bigger cage. But by day 10 he was having great fun, climbing and exploring and just looking happy!
I think you'll find you have a different hamster when you have her new cage set up and she's in it for a week or two - the amount of space really affects their behaviour.
I found this video by Erin's Hamsters was a Godsend - tips for cage cleaning - it is one of the things that many owners do without realising how much it stresses their hamster - after watching this I now do the cleaning in stages and never remove all of the substrate and leave the bottom 2/3 and just change the top 2/3. I think you'll find it helpful for now and when she has the new cage.
https://www.youtube.com/watch?v=5jVnEjrXbww