Hi there,
Congrats on the big move. As for advice, particularly on where to live, it really boils down to what you're looking for and your budget. I'm assuming since Chelsea has been recommended, that money is no issue for you

Chelsea is lovely but I'm not sure what makes it particularly good for Americans. Some very nice, accessible areas are Marylebone and Kensington, (Marylebone, is, for all intents and purposes, central London and Kensington is well-appointed with plenty of tube and bus links). But what are you looking for: green and leafy? In that case, perhaps Hampstead, Primrose Hill, Belsize Park as they're all bordering the Heath. Bustling? In that case, maybe Camden or Angel. Edgy? I'm too old to know what's edgy anymore, but once upon a time it was Brixton or Hackney. I'm sure others could fill you in.
Maybe the best advice would be to see if you can get your company to put you up in a hotel for a little while to give you an opportunity to explore a bit and see what works for you. London is huge with lots of different neighbourhoods.
Also, I would try to get your company to arrange to open a bank account on your behalf. I found opening a bank account here a slight hassle before receiving my first paycheck but my company helped everything along so that I could get my flat sorted out. Basically, if you don't have a bank account you can't get a flat but if you don't have an address in the UK, you can't open a bank account. You see the predicament?
Some annoyances I experienced: getting my first credit card that I was to use for work-related expenses to be reimbursed and finding they would only give me a £250 line of credit. Not being able to get a phone contract because I had no UK credit history. Work eventually sorted these things out for me but you may want to bring these things up.