Story of my life. BR London flights arrive in TPE too late to connect with anything and depart too early in order for you to have come from anywhere else.
You might want to consider if it's really worth it. HKG-TPE-BKK-LHR is quite a long way around especially given the overnight.
I buy SGN-TPE-BKK-LHR (an equally "inefficient" journey!) and recently have had to pay as much as double that. The ticket now never includes layover accommodation, though it did once. It's OK for me because I always want to stopover. So unless the fare very specifically offers accommodation, I doubt whether you will get it.
The More lounge is fine, you can snooze there. But it's not open overnight.
The "Evergreen Transit Hotel" is probably the most convenient solution or landside just outside is the Novotel. You can google both to get an idea.
Another possibility is if you could arrive from HKG a bit earlier you can at least have your night in Taipei. Do something in the evening. But you really need to be out of the hotel by about 6.30am to make BR67.