If you can get to the Piccadilly line platform in 45 minutes or less, you should take the Piccadilly line train at 1347 which arrives Kings Cross St Pancras at 1449. Costs £3.10
If you just missed that, you can get the 1352 TfL Rail stopping service (from the rail station not the tube) to Paddington which arrives at 1426. At Paddington, from the rear end of the train, go up to the footbridge, turn right along the bridge and walk to a concourse. On your right is the taxi rank. On your left is the Hammersmith and City line station which has trains to Kings Cross St Pancras without needing to change. There is a tube at 1434 which arrives 1447. Costs £12.50
If you arrive landside later than 1350, get the 1411 TfL Rail service to Terminals 2/3 then change to the Heathrow Express at 1418 which arrives Paddington 1435. The H&C line runs at 1439 arriving 1452 or 1444 arriving 1456½. Costs £24.40. A taxi is unlikely to get you there faster than that, unless the tube has delays.
Thank you for taking the time out and giving such detailed instructions. Haven't gone through LHR T4 in a while. If I get off the airplane at 115 what time can I expect to make it to the Rail station?