London Layover

I would stay at Paddington rather than get on the Tube with your luggage. There are lots of hotels to choose from right close to the station entrance. For something with local flavor, not a chain, take a look at the Pilgrm (yes, that is the correct spelling):

https://thepilgrm.com/

Or for a more upscale chain hotel, we like Hotel Indigo. If you like Hiltons, there is one right there as well.

With any of these, you can be checked in and luggage-free within a few minutes of arriving at Paddington on the Heathrow Express. Then you can head out to explore the dinner options close by (you can get suggestions here).

Next morning, check out and leave your luggage at the hotel while you set out to see London highlights. You can return to the station and take the Tube anywhere you like, or walk 10 minutes down to Bayswater Rd and ride the 148 bus right to Westminster Abbey to start your exploration. The bus will take longer, but you will get a sightseeing tour on the way.

https://content.tfl.gov.uk/bus-route-maps/key-bus-routes-in-central-london.pdf

I would suggest Premier Inn Paddington which is about a 5 minute walk north of Paddington Train Station. The Premier Inn is a very popular hotel chain in the U.K., and this is a very nice property with its own restaurant. There are also a bunch of inexpensive restaurants across the street from the train station.

The underground station can be accessed from the train station or at its own entrance across the street and down the block.

You've kind of put the cart before the horse here in booking HEX tickets before knowing your hotel. It would have been better to book a hotel, then decide whether to take tube, HEX or Elizabeth line into London, based on which got closest.
But that is done now. On that basis if your flight is on time it will be 9.30pm at best arriving at your hotel, so just find one near Paddington.
Next day you need to be back at LHR by 1.15pm, so leaving Paddington by 12.15.

Boat tours certainly run all year. But the Thames Clipper/Uber Boat runs late at night. To see London by night from the river the last clipper leaves Embankment pier at 2232 and Westminster at 2236, all piers to Greenwich 2325, North Greenwich 2334 and Barking 2352. For instance from North Greenwich there will still be last Jubilee line trains back and also the all night bus #188 (from memory) back to Tottenham Court Road via Waterloo. The first eastbound Uber boat next day from Embankment is as early as 0718.
Going in the other direction, towards Putney there are boats as early as 0616 from Embankment- for tube, train or bus back.
There are lots of scheduled Red London buses which run until midnight, some all night- some routes run through central London so are great sightseeing routes in themselves.
One that has cropped up recently on another thread is the #139 Waterloo to Golders Green- that runs 24/7. The #205 Paddington to Whitechapel is also 24/7- ok that is a 1/2 circle round the northern edge so is not hugely scenic but is very useful
But the 11 Fulham to Waterloo, the 15 , 25 and the 38 (all scheduled bus routes which double as sightseeing routes) are all 24/7- better and faster than the HoHo routes.

The #274 mentioned by Lola does NOT get you to Westminster- that ends at Lancaster gate- you want the 23 down to Trafalgar Square and Aldwych then switch buses or walk- that is another 24/7 service.
The 23 connects with the 38 and the 390 at Hyde Park corner for Victoria.

You will need to be back at Heathrow 3 hours before your departure time so in reality you will have only the morning for your Quick Look at London. Make sure you check in on line in the 24 hours before departure from LHR to avoid the queues at the check-in desk. Also, though your arrival flight is scheduled to arrive at 8.30 pm, it can take at least an hour to deplane, get through immigration, and on to the Heathrow Express…..so it will be at least 10 pm when you arrive at your hotel.

Correction—the bus I meant is the 148, which runs along Bayswater Road to Marble Arch and then turns to head to Westminster Abbey and London Tower. We have ridden it in both directions, and it is a nice scenic route.

Thank to isn31c for the correction.

Possibly Windsor would be a good short stop, castle in the am, lunch, walk around and head back to your hotel.