Is the angle to just raise some money for a deposit (a rental deposit? or are you looking to buy?) or to use this as a bit of an outrageous stunt to expose your landlord by name and shame them into either fixing what's wrong with the property or something else?
I know it's very hard to move while renting when there are pets in the mix but what was your long term plan with the pets situation?
Can you move? I'm guessing you're in a private rental from what you've said.
If you're able to move, start exposing the landlord's disregard for your health and the state that the house is in if it's, as you say, top to bottom.
In the meantime, get a decent dehumidifier or two, make sure you're airing the house, wiping moisture from windows to limit the damp, move furniture away from walls so air can circulate, open windows in bathrooms.
Do you work from home? If so, maybe it's worth using a warm space or a local library during the day so you're not in your damp home the whole time.
I'm not sure how the damp affects the diabetes. I know keeping a home warm is important for blood flow for diabetics. But I'm not a doctor, nor do I have diabetes.