If you are going to play on then you may as well play on to mate (some beginners want a chance to get the killer knight/killer bishop/etc. achievements).
If your opponent is a kid under 500 there is a good chance (maybe even 50%) that K+Q vs K will be drawn and a better chance (maybe even 90%) that K+2Q vs K will be stalemate.
You can take your time analyzing but don't just waste both players' time delaying your moves just to annoy the other person.
I've never played an arena event but from what I've heard the key is to play (and preferably win) as many games as possible in a fixed period of time and playing on in an obviously lost position would reduce the tournament scoring opportunities of both you and your opponent.
Play until the end! You might even win or atleast get a draw. So I suggest to not resign.