The problem with Heathrow Express is that although the trains travel quite fast it all goes to waste because they dump everyone at Paddington, and then you have to find a way to finish the journey - causing a delay which negates their premium fares. There are many taxis at Paddington but the ride to the hotel will take around 20 minutes and cost around £15 on top of the train fare.
Check your hotel booking, this is the Hub in Dacre St, postcode SW1H 0DJ? It's about 1 minute walk from St James Park Underground station. Put the postcode into Google Maps, drop the Streeview icon half way along Dacre St . Turn it so that the building site is on the left, the hotel is on the right and you should see the green logo at the door. Now you know where that is, click to the end of the street, facing you is Starbucks, turn slightly to the right and there is the station. Adult cash fare from Heathrow is £6 each, journey time is only a few minutes slower than HX + taxi.
There is only one Underground line at Heathrow and all trains go the same way. You need to change trains, normally this is not a problem if you do it at Barons Court, cross to other side of platform for east bound District line. Wait for atrain via Tower Hill (probably to Upminster), definately NOT Edgware Road. Easy to do, just read the front of the train or the screens. Unfortunately if the day is 7th October you can't because there is engineering work on this section of District Line, in which case you change a few minutes later at South Kensington, which involves a change of platform to east bound Circle/District but all those trains will be heading your way.
At St James Park station you need the exit towards the front of the train (Broadway) and at street level the door on the south side (opposite side to the actual park).
The bad news is that even if the plane lands on time, you are unlikely to get through Immigration and Baggage reclaim and get out to the ground transport before 08:45-09:00. The journey to the hotel will take an hour (Heathrow is WAY out of town). You would need lottery winning luck to reach Parliament by 10:20.