I looked into your match history, you can't really blame this on towers being weak. Your team gave tower first blood at 18 minutes, that's pretty late (in my experience, it happens between 8 and 13). Vayne had a completed item and your tower might have already taken minion damage.
Also, despite all your kills, it seems to me that you had very low pressure in the bot lane. With such a lead you should have been able to secure the first tower, a drake or at least deny farm. It might not be your fault, I don't know the details of your game. Maybe you couldn't afford to push because you had no vision on mid/jgl and Kha couldn't help because he was so behind or something else...
Anyway, I think the main reason you lost is because you had no reliable CC in teamfights while Leona, Cho and Syndra can lock down at target and peel for Vayne.