What about the hotel itself? Some hotels I've stayed in have had Sky Sports in the rooms and also it may be playing in one of their bars.

It's worth asking about both these points. It's specifically worth asking about the event itself, for example they may show the FA Cup final but not other league or cup events. So if someone says they've seen Sky Sports on at the hotel bar before they might not in your case.

Otherwise why not go into London or Brighton and find a pub there. It's very easy to get back to LGW (or GTW in rail-land).