While you can't checkin without your sister, you might well be able to leave bags there in advance.

I'd suggest you get an Oyster card at Heathrow, and use the tube to get to the hotel. Use the TFL journey planner to check what is actually the best station to use. Leave your bags at the hotel if you can, and go explore. If not, decide where you want to sightsee, and find the nearest mainline station to there. Take the bus, drop your big bag in left luggage (at about 8 quid per bag for up to a day), then sightsee and collect later