<6 month living in UK

I am a US citizen who wishes to temporarily relocate to the UK for just less than 6 months. My Russian girlfriend will be entering University for a 12 month program starting in October and I'd like to live with her for 5.5 months starting in October.

• I am 31yo white male.
  I do not have any recent British ancestors.
  I do not own a house or land; I will move back with parents while over in the UK to minimize expenses.
  I am a self-employed web designer with less than £10k in savings. I have worked full-time in my past for a couple of years but am currently freelancing.
  I hold a bachelor's degree from an American university.

For a five and a half month visit where I strictly live with my gf in her flat, how much in funds do I need in my bank account? I can only provide bank statements of my income in the states from clients that I get on a regular basis.

I also will need to work on my projects for my American clients while I live in the UK so that I can wire money and support myself, pay for food and living expenses and help contribute to my lifestyle. From what I've read, it's not legal for me to work while in the UK. But does this apply if I'm still doing business for my American clients?

Thank you for your advice and replies!

Please don't multiple posts on the same topic.