HI Resortini,
I hope it is not too late for a reply, but I went through same situation with my under 18's nephew. I suggested The Normandie Hotel ins Sussex Gardens,London,W2. It is very close to Paddington tube station, where you can get to Imperial College in less the 20 minutes. It costs £35-45 per night. Another option is to search for a hostel at yha.org.uk and chose St Pauls one. Its very central , very clean and cheap,around £10-15 per night.
Good Luck