how much raise rent - also when tenant won't communicate

Tenancy Types and Management

My tenant has not answered my calls for over a month. I had issued a section 21 notice 5 months ago and she is looking for somewhere new to live but not found anywhere. She tends to ignore problems. I wanted to do up the flat and rent it out through a management co and concurrently sell it. I wanted to do this before I adopt 2 little girls who are from that area, so I can no longer manage the flat myself. She has not moved out and the children are about to move in so I no longer have the time to do the flat up anyway and I need to do something about the fact that I can't get rid of this tenant. I am barely covering my costs for the flat and am worried about having a big major works bill coming up. I have not put up the rent in 4 years and it is £120 below market rent, which housing benefit are paying.
My new option is to raise her rent. But she is not responding to my calls. How do I raise the rent? I intend to raise it to £800. This is still below market rate - but can I raise it from £730 to £850 in one go? Can I do this directly with Housing benefit as they are paying me directly? does this automatically mean a new tenancy agreement? what if I send her one and she doesn't sign it? Any advice please about raising rent, or starting a new agreement. (there is an advantage to me starting a new agreement which is then I can make sure I do it right and can evict her in six months time).