you have no written agreement. That means you still have a tenancy, you paid rent and he/she accepted it.
bad news: landlord can raise rent by as much as he/she likes
good news; if you don't want to accept that, they must give you 2 months notice to expire on the day the rent is normally paid. End of. The law. The notice must be in the correct written legal form.
number and age of children irrelevant. Your landlord wants you out so is using this rent rise to get you out. Do you really want to keep paying money to this person long-term?
you don't have to leave on the day the notice expires; you can wait to be evicted. This will obviously have effects on your future suitability as a tenant. Before you start considering this, speak to your council to see what their policy is.
get house hunting now. For the next place, ensure that the deposit is protected, there is a gas safe cert, and that you have a written agreement for the first six months at least. If you can get longer, do it. Look to rent a buy-to-let not someone's home, that way they won't be wanting to move back in. Be an informed tenant.
don't rent a dump, they don't improve. And don't rent from a 'friend'.