Interesting. I've never lived in Hammersmith and never will. My new flat is actually in a very different location than where I've been before. I'm now in the centralist of central london.
wc1a 2pj london - Google Maps
London has 5 major airports:
Heathrow
Stansted
Gatwick
City
Luton
The only one I actively avoid is Luton. 90% chance you fly into Heathrow anyway.
British Airways has always been my choice airline but I'm told Virgin Atlantic may be nicer (have never flown with them). Chicago has probably 20 non stop flights to London daily, Indianapolis has a whopping 0 so I recommend you book from Chicago.
Everything in London is accessible from my flat.