There really isn't a right and wrong answer to this just different options but IMHO I'd stay at LGW over night rather than go to LHR.

You would (and possibly more importantly your kids) be in bed at least an hour sooner than if you made your way to LHR

BTW that price from Farringdon to LHR is for the Elizabeth (more informally Lizzie) Line whiich is a train service

The actual tube would be £5.60 per adult if using a contactless card and cheaper for the kids but adds an extra change of line into the mix plus is longer. But if you wanted to use the tube then I'd change at Blackfriars for the District line then later to the Picadilly.

According to the timetable the last Lizzie to LHR from Farringdon departs at 23.15. Which would be impossible for you to catch with a 10.30 arrival at LGW.

There is also a large gap in the National Express coach.picking a random date there is a coach at 23.30 and the next is 03.35. If you do use the coach (at whatever time) make sure you check the luggage sizes and amounts and book accordingly as paying for extra baggage on the day is more expensive than in advance)