The contract is due to end in the Autumn
What do you mean by this? In your op, you say you’re on a rolling contract. For me a rolling contract means you haven’t got a current fixed term extension on the AST and that you are on a Periodic tenancy. But then you go on to say the contract is up in the autumn.
If you’ve signed a fixed term extension to the AST, the ll cannot increase the rent until that expires, which you indicate is in the Autumn but the ll says is in April. If you’re on a Periodic tenancy, ie no fixed term extension, and presuming you pay monthly, the ll can just give you a month’s notice. The anniversary of when the AST expired would be irrelevant. But the month’s notice is a minimum as if the ll tell you the day after the rental due date, that then becomes 2 months minus a day. As others have said above, you can dispute the section 13. And the ll can only increase the rent in any 12 months period.
The improvements you’re being asked to do are batshit and the fact that the ll has sacked the tenants indicates to me that they’re utterly clueless... and / or see you as a cash cow. They need to give you 6 months notice currently to leave but that may change at the end of March and revert to 2 months. My thought is this is unlikely. But tbc.
Have you checked what the market rate for the property is?
You talk of a damp property. It sounds as though your company has paid for a new kitchen and integrated appliances. It was very unwise to allow money to be spent on a damp property. You would have been better to have waited to buy your own property if that was on the table - or maybe you’re on an expat contract / there on a short term basis.
My take would be to find somewhere else to live and without the batshit lls. The kitchen is a sunk cost. You likely can find a nicer property without the damp issues and with a nice kitchen. It may cost you the same as your rent increase. But at least you’d be away from the ll.
It is very unusual to change kitchens in England due to the short length of tenancy agreements.