That is a good point by Amethyst Ice - rather than my "kill them" approach! Which probably came from trying to battle a mouse problem for a long time. Wood mice or field mice are quite sweet looking little things and quite small. A house mice has more of a pointy face look and normal mouse size. I did do the humane traps at first but after a week of getting up at 6am and driving a mile I was worn out and just set traps and there were dozens of them drawn out by the bait.
Catch the first one and see what happens
Your hamster's cage should be fine if it's a bin cage with 5mm mesh. I would put any hamster food in a sealed box though. And also bedding if possible.
Do you live in a rural area at all? Or an older place with floorboards? I had them once before in a modern house - I had a large back of black sunflower seeds and an old mattress in the garage and it had attracted them in - they had a field day with the sunflower seeds and the mattress! Unfortunately they then got into the house roof space via pipes with lagging round that went up through the garage roof.
If there is a nest then it is quite important to remove all the mice somehow as they can chew through cables.