Do you have a picture of the other cage - the one with tubes in? We might be able to recommend some way of re-jigging it - depending on how much bigger it is. Mollie is correct about minimum welfare cage sizes. It's about 80 x 50 for a Syrian and 70 x 40 for a single dwarf hamster in the Uk. In the US it's 360 square inches (just the floor area - width x length) which is quite a bit smaller. An in Germany it is 100 x 50cm as a minimum. These measurements are for continuous floor space in one cage. They are quite active little things and need space to explore and do hamstery things, and once you have toys, house and wheel in that can take up some of the space too.
The cage on the photo does look very small. How big is the one with the tubes? Unfortunately pet shops do sell these small cages to people (we started out with something way too small). They also find it quite stressful changing cages though, so best not to swap her between them just yet if she has already been moved once.
Some people make bin cages. This Ikea Samla bin is a good size for it - but it needs a bit of diy for ventilation. A panel cutting out of the front and top and mesh putting in the gap and held in with cable ties. Do you think your Dad would be interested in helping you with a diy project? The cage she is in now would be good as a play cage for when cleaning her out. Anyway, if you have a photo of the cage with tubes we might be able to advise on how to block off the tube holes (you can use one or two of the tubes as floor toys maybe).
SAMLA Box - IKEA
It's difficult when parents think somehing is ok and don't want to change it, but people have taken cages back for refunds shortly after buying them, saying it is too small and not suitable.
Amazon in Gemny also as a good seection of suitable cages. This one is 100cm with small bar spacing and not too high and not too expensive.
Mäuse- & Hamsterkäfig ELMO "XXL": Amazon.de: Haustier
There are also pleny of 70 to 80cm cages on there as well, but lets look at your tube cage first.