Increasing the rent

I plan to increase the rent on a tenancy AST to cover the new discretionary licence costs foisted on us by the local council. The AST is is two+ years old, originally on a six month AST it has rolled over monthly ever since.
What notuce period do I need to give if using a S13 notice - is it six months or one month?
I have an inclination to give them a S21 notice and offer a brand new tenancy at the new rent and re-check their credit rating, employment status etc. within that as I have an inkling that all is not well between the two tenants, so this may force things out into the open. Would that be a legal and valid way forward? Thanks