Throwing a bit of cash as the problem can simplify it ! The Sofitel is connected to the landside area of your Heathrow terminal. They offer day-rates in the region of $200+, though prices can go crazy. It might be worth checking overnight rates: perversely these can be cheaper, and by the time you get to the hotel it will be within the generous interpretation of check-in times, and the overnight rate would allow you to stay beyond day-rate limit hour.

There are, of course, many other hotels around Heathrow with much lower rates: but none offer the convenience of the Sofitel




https://all.accor.com/hotels-service...index.en.shtml

https://group.accor.com/en



The opportunity to get some proper bed-rest and and your very own bathroom might be welcome.

You'd need to go through immigration to get there, and go through the security business when you return airside.