This is me. I cannot switch off, ever. I'm part time freelance in the Arts, part time university teacher - I could spend my life researching and i'm constantly bloody thinking. I can't read fiction anymore or watch TV. After 15 minutes I'll think..."I'll just look into x y z", do a batch cook, Marie Kondo my room etc etc". Always have to be doing!
Long term, you need to start meditating. It sounds like you are wired and your brain is not used to taking a break. You need to teach it how. Start off with 5 minutes...YouTube and Headspace are great. I started off with 30 minute guided meditations and frequently fell asleep which isn't ideal. Also try and avoid tech as much as possible. Put your mobile and devices in a different room or better yet, let them go dead for a few days.
I really find those adult colouring books relaxing. Maybe because i'm doing something but part of my brain gets to switch off. Yoga is also a Godsend. Yoga With Adriene on YouTube has videos for when you are ill. If you do the Yoga and Meditation first it takes care off the rest. Also, playing relaxing music on Spotify is good. I like the Coffee Shop Playlist. You can get an account for free if you don't mind listening to the ads and listening on shuffle.
For your chest infection, try and get lots of steam into you but don't use a lot of mentholated products as they only dry you out further. Popping some thyme into the bowl of hot water is good. If you aren't vegetarian try and make (or get someone to make you) chicken soup made with a chicken carcass or bone in chicken, aka Jewish Penicillin. Plenty of tea and hot drinks. Don't use cough sweets (again they will dry you out), any boiled sweet is better. Pineapple juice is great, also honey and ginger. Once you feel up to it and the weather is ok, wrap up warm and go for gentle walks. 15 minutes twice a day is great as and when you are able. You need to rest in bed sometimes but not too many hours at a time as this won't help the infection in your chest to clear, especially if you are lying flat.
I really hope you get better soon!