If you are really desperate then you need to just move her in to a bin cage with a decent sized wheel. It is no hardship and she will soon adjust.
If you want to keep her in the cage then a simple solution for you to get some sleep is to go to Boots and buy some ear plugs. The mouldable wax ones you get for swimming are quite comfy. Do you have a large wardrobe or cupboard she could go in at night? Needs must and you need your rest.
Try and block off the area she is chewing. Get a plastic folder or a log bridge and make holes in it and wire it over the chew spot. Thing is she will likely chew on the log bridge which will also be noisy as is wheeling so having her in the bedroom when you are a light sleeper is not the greatest idea. Could she go in say the kitchen once everyone is in bed at night? Drastic measures are needed to solve this if it has become so bad you are thinking of rehoming.
No hamster will be silent in a cage though as just their normal nocturnal activity makes some noise remember. My best advice is to negotiate another room space for her at night
You are unlikely to get a large enough tank cage in P@H. Are you near a Jolleyes?