How to get to London when arriving at Heathrow at 10:45 PM

There are hotels around Heathrow. I would be included to stay in the area and take the tube in the morning. If using the tube it will be about an hour and half getting to Eurostar. You need to arrive at least 30 mins to go through security prior to boarding the train. Another option which would be about a half hour or so quicker and more expensive is the Heathrow Express to Paddington station and then the tube to the Eurostar or a local cab.

Frank suggested the Heathrow Express as a possibility, and I concur in that. My wife and I arrived about the same time and had no difficulty catching the Express to Paddington. There are several hotels around Paddington, so you could easily walk from the station, check in, and the next morning take a taxi over to St Pancras....but I would be inclined to take a taxi to a hotel close to St Pancras to make the morning stress-free. You can search websites of the various chain hotels in the area (use TripAdvisor's map centered on the station to identify them) to see if any are running last minute specials. Look for Accor (Ibis, Mercure, and Novotel hotels), Holiday Inn, Premier among others.

Part of the problem is that the last H Express run is right around 11:45 while you have an hour it could get a little tight if you plane was late. I would hate to be committed to hotel around Paddington at that time night. I think there is a Yotel hotel in Hearthrow that charges by the hour. Catch six hours of sleep.

Hotel Ibis Euston is about a 12 minute walk to St. Pancras.
We stayed there to have easy access to the Eurostar.