You can't pay LESS rent without going into arrears. If I were your landlord I'd be evicting at the next opportunity. (probably) You need to discuss this with your landlord like a grown up. Explain politely the impact on your health. And ask him to help out.
Ideas for a free shower -
1) Landlord pays for you to go to a pay as you go gym.
2) You use the shower at work
3) You use a kettle and sink, and a flannel. I was 30 years old before I had a shower in a house I lived in. Parents didn't like to waste money filling the bath. I believe it's called a strip wash. Very old fashioned.
It's summer - eat salad. It's cheaper than eating out.
When our boiler went, it was minus 10 outside, and it took a week to be sorted. The house was like a ice block, and I was £2K poorer, so thank your lucky stars that it won't cost you much, and will be fixed within 3 days.
A new boiler in one of my BTL's would take 40% of the years profit. Whilst my insurance covers putting the tenant up if the house is uninhabitable, I don't think the loss adjuster would think a boiler out of action should result in the house being uninhabitable.