Are you travelling on a Sunday? Normally, the journey time from King's Cross to Newcastle is a little uner 3 hours, which makes it far more attractive a way to travel (particularly with good value First Class fares and onboard Wi-Fi).

(Oh, unless you're staying at Heathrow anyway and need to get over to King's Cross for the train...)

In Newcastle itself, you're well served by public transport - the Metro system is excellent. I love the city - I grew up there, and it's got a fantastic spirit, and more attractions that you can shake a stick at.