Usually, at this point:
Spoiler :
As for your game, you can try and buy a few friendly AI's votes; you mentioned being on good terms with them. Just send your diplomats to their capitals (as diplomats, not spies! And well before the UN voting - it takes 6 or 7 turns of for them to settle in). If an AI agrees to accept the trade, it gives you 4 votes. The price may be arm and leg, but if those votes give you victory, it is not too much. It is better to bribe the AIs first and only then ally the few lacking CSs, otherwise the AI may 'sense' the danger and just refuse any bribe from you, however friendly it is - I've had such an experience. Also, get Globalization, if it is within reach.
Yes, I think I may have approached it the wrong way around - I've almost got all of the City States on board, yet the AIs are refusing to budge - even the ones who share my religion will not give me their votes. I think I may just get Globalisation and that should be enough to swing it.