Event at Wembley stadium

I think you have probably long missed the boat for booking decent local accommodation a reasonable price. As an example, the Hilton close by is roughly ten times the price for a room on 19 July as it is for 20 July and many hotels are completely sold out. Even the absolute dives remaining are £200+.

So pick somewhere for the whole stay close to the night tube network. Wembley clears out by public transport in under 90 minutes but with the time of this event you also need to consider the full journey. What is your budget?

If you look on google maps, you will see a number of hotels around the stadium and also quite a few restaurants.
Check availability on Booking.com, or direct for Premier Inn, Ibis etc.
You may be looking quite late.
These are all there to service the stadium - it's not a picturesque area at all. But as safe to walk round as anywhere..

6 adults? That is most likely 3 rooms.

If you are having trouble finding available accommodation near the stadium (there are a lot of options but I understand they do book up), I would suggest staying in/around Marylebone for your whole visit. There's a direct train from the station called Wembley Stadium to Marylebone and it's a nice area. It's also easy to get from there to St Pancras for your train to Paris (if you are taking the train). To be honest there's not a lot of places in London you can get snacks/drinks late at night... most things tend to close around 11pm. But there are lots of nice pubs around Marylebone for a slightly less late night drink. Good luck!