How can we convince my uncle to enter a nursing home?

So sorry that you're in this tough situation! From a legal standpoint, if your uncle has been deemed incompetent, and you are his guardian, you can decide what's best for him. This means if you feel the best place for him is in a nursing home, then this is where he should go. However, you sound conflicted about this decision. I'd speak to the social worker again to see if it's feasible for you to keep your uncle at home. I don’t know your uncle's financial situation, but if he has the means he may be able to stay at home with 24 hour care. But keep in mind that this is expensive, especially over time. His finances may not last. There are programs in some areas of the country that help people live independently, providing day care and home care. Ask the social worker if she knows of anything in your uncle’s area. I would also consider his medical illnesses. If he's really ill (and it sounds like he is, with your mention of hospice), then home care may not be possible.

From my experience, the elderly don't want to move away from home, as going to a new place is very difficult. Something that might his transition easier is touring facilities in your areas to see which places are a good match for him. Understand that he will be angry for at least a while and this is normal. As he gets used to his new surroundings, you may see that his health is benefiting from more supervised care. Remember, his anger is really not about you placing him in a nursing home. It's about his losing his independence, which isn't your fault. Please don't feel guilty about trying to keep him safe and comfortable. You're trying your hardest in a difficult situation.