Dont cycle, tube is good. It is a 5 minute walk from tower hill to the hotel and an easy walk to wembley from the station.
The hotel may seem like it is in a bit of a dead zone, which it is, but you are a short walk from some nice places. I work nearby.
Get down to St Katherine docks, see the bridge and the Tower. Cross the river to see hms Belfast and the Mayors office. Walk down Shad thames to the design museum. Maybe walk round the financial district to see the tall buildings up close. Maybe walk along the southbank to the tate modern and the globe, then cross over the millennium bridge to St. Pauls.
London is a great place. if you want to eat/drink cheaply there are a couple of tesco metros nearby to pick up basic supplies at sensible prices.