I lived between London and Glasgow for a while running my business, although I'm pretty much back in Glasgow now.
What I found about working down there is that whilst the facilities are superb (British Library is one of my fave places ever!), I'd not recommend it for start ups. The reason being you are one of 7 million people, so there's more competition for press, funding, space, limited resources etc.
And if you are thinking of expanding and hiring staff the costs are extortionate and they don't stay long because no one really wants to live in Lond, especially not on an admin salary - it's not fun, I did it for 5 years prior to starting my business!
Presuming you aren't either starting up or hiring, there's still an element of wear and tear on you too. For example, if you are working from home (and that is most likely since office space is extortionate) the isolation and cabin fever really does get to you. Plus everything takes about 3 times as long, which for an entrepreneur whose time is money, is a disaster. E.g. Supermarket shop here can be done in 20 mins because I have a car, there's parking, I do it once a week because I can load it up. Supermarket shopping in London has to be done in dribs and drabs since you mostly wouldn't have a car in Central London and it takes maybe 10-15 mins walk each way, never mind the shopping. Same thing for seeing friends - in Glasgow it take me a maximum of 20 mins to get anywhere, but in London to visit friends I might have to travel up to an hour to meet up - you're losing time and so you make the effort less often, and become isolated pretty quickly.
What really broke the camel's back though was house-sitting a huge house in Fulham for the summer. I had 5 bedrooms, a garden, a cleaner, a car - right next to the park and the lazy River Thames. Realistically this was the best the London could offer its residents. I still found it deeply tedious and decided to get out - never regretted the decision!
In short, to entice me back to London I would need to be being paid at least £30k more than I am at the moment in order to have the same quality of life I have here. Up to you though.... Sometimes you have to live the reality for yourself.
A better idea would be to commit to being in London maybe 2 days per month - make all your meetings during those two days, stay in a nice hotel where you can host people for meetings, attend some networking events... But it'll still work out cheaper and more effective than living there.