From the Shelter website:
A court will refuse to order your eviction if all of these apply:
you complained to your landlord or letting agent in writing before you received a section 21 notice
you complained to the local council because your landlord didn't take steps to fix the problem
the council sent your landlord an improvement or emergency work notice
A section 21 notice served between the time of your written complaint and the council issuing an improvement or emergency work notice will also be treated as invalid."
I'd suggest that you get a letter into your landlady pdq, ideally send it "signed for" so you have proof of delivery and keep a copy. That way, she can't claim you're making it up!
If the Shelter website is right (and they're very reputable), she won't be able to evict you.
Also contact your council's environmental health department and explain to them that your landlady is refusing to carry out essential repairs. They will come and check it out, and then try and get the landlady to do the works.
Check that when you got your tenancy your deposit was placed in a deposit protection scheme, that you were issued with the "How to Rent" booklet and that there was a gas safety certificate in force when the tenancy started. Failure to provide any of the above means that the landlady can't legally issue a S21 notice (notice to quit).
The council should also be able to advise you. They now have a statutory duty with regard to homeless prevention, and should have staff who can give you advice. Ring them and explain that your landlady has said she will have to evict you.
Please try not to let this worry you. On the face of it, it sounds as though she won't be able to evict you if you have made a written request for repairs to be carried out, but the council will be able to give you a definite answer.
Some landlords really take the piss. She's been happy to take your rent, but claims she can't afford to fix your plumbing and electrics, both of which can be potentially dangerous if faulty. She has no business letting out a property if she can't afford to keep it in good order.