Who is responsible for the boiler?

You are. Heating and hot water are on the landlord’s side of Section 11 and cannot be moved to the tenant by the agreement.

That covers repair and replacement. If the boiler is beyond economic repair, replacing it is your cost, not something you can defer indefinitely.

The tenant is responsible for using it sensibly and for reporting a fault promptly. Failing to report something that then gets worse can affect what they recover, but it does not remove your duty.

An annual service is not legally required in the way the gas safety check is, but it is what keeps a boiler from failing in December.