I've been following your thread and I don't really have a lot to add.
However I think going back to your GP might help. As others have said there is a chemical imbalance in your brain and tablets might help, maybe you need something stronger. Did your GP refer you to counselling, if so please follow this up, this can take time to be accessed on the NHS and maybe consider paying for some private counselling.
Whatever you think about what you read of what others go through, you have been through an awful lot whilst caring, and you did a brilliant job for your wife.
I don't know whether the experience I am going to recount below will help, but it might:
My son is dyslexic, and for about 6 or 7 years I did battle with the school system to get him help, on and off I couldn't sleep for being so upset I hadn't got him help, and that I had failed him. I had some counselling last year, and the counsellor said to me I didn't fail him as I kept asking for help, I did all the right things however those I had asked didn't help. This gave be a big mental release with someone else telling me it wasn't my fault.
It isn't your fault your wife is how she is, it isn't your wife's fault, it is the disease's fault. You did your very best to keep your wife at home, and you are doing your very best for your wife now she is in a care home. She needs you as her advocate in the care home, and to navigate the many hiccups in the care system that she yet might come across.
It isn't easy to go on do things when you feel as you do, and the first second or third time you do them they won't make you feel any better, in fact you will be thinking I shouldn't be able to do this, as life shouldn't have turned out this way, but you will be taking something out of doing them. I lost my first child (stillborn), which I know is very different, so I went back to what I did at the time, cycling and canoeing, but as I did them all I could think was I shouldn't be doing this I should have a baby at home to care for. But slowly slowly I took pleasure in doing them. This might not be right for you but for me I'd suggest try and find something out of the house to do every day - it might be something you did before or something new. It might be that one week you do something on Monday, the next week Monday and Wednesday - slowly build up. If you can find something that will involve interaction with others - however disconnected you feel - I think this will help you slowly reconnect. One day drive/ walk to a local cafe and buy a cup of coffee, chat to the waitress and ask them if they are busy today, you will slowly become a regular and chat with you. Go to a local park and walk round, I like looking at the light on the leaves of the trees, it changes all the time, and then when the trees are bare the light is different. If you like dogs see if there is a local animal rescue centre in need of a dog walker, they are normally desperate. You mentioned a bike, is there a nice traffic free route near you that you can ride, take time to look at the plants trees on it, this might help distract your thoughts. Everything you try to do, try and do 4 weeks running as it gives your body and brain time to accept it.
This might all be too much, this might not be who you are, but hopefully there are some thoughts here that connect with you to help you.