Putting it bluntly, had he left at the time Sweden wanted him, he'd be a free man by now. Have you seen the prisons in Sweden, he'd have more freedom there that he does in London.
Yes, but then he'd most likely be a convicted rapist or other flavor of sex offender. That'd rather undermine what he perceives as his credibility. Having made up the lie of hiding from American aggression and using it as an excuse to duck the Swedish investigation, he pretty much can't come out, at least unless we grant him safe passage to a non-extradition country, which we obviously won't do.
It's clear Ecuador is getting a little more than pissed off as well since they are now giving him rules that he must adhere to.
I assume if they had any idea he'd still be in the embassy all these years later, they'd have turned his request down flat.
Ultimately, the best way to make sure he leaves is to simply deny him the oxygen of publicity. He seems to me to be a shallow megalomaniac with an ego the size of a small planet (or spacestation). Denying him external communication is a good way to reduce that oxygen. He'll never leave otherwise.
There is no possible future in which either the UK grants safe passage to a non-extradition country without leaving the embassy and being arrested for bail jumping, and there is no possibility of an American president granting him immunity such that he feels safe from extradition. The irony being, of course, that Sweden was a lot less likely to extradite him than we are - see the "Natwest three" for details. Eventually, he's going to just have to leave and face the music - the only real question is how much he's willing to make his children suffer before he does, or whether he dies of cancer or something beforehand.