My take:
Jordan's king can't get too involved in Syria without exposing his own reign to the local Jihadis in his own country, particularly with a ground force. Staying in power and avoiding the bayonet up the ass are further up the ladder in his mind. Like most rulers / governments in the world the king knows that he and his country stand alone and can't count on Comrade Obama to come to his defense.
Israel is more likely to help out with airstrikes and intel than the U.S. is. Both Israel and Jordan have a deadly common enemy.
To back up above statements:
Jordan Unlikely to Become a Regional Sparta - David Schenker (Foreign Affairs)
For the past six months, opposition in Jordan to the war by the anti-ISIS coalition was broad-based. However, that changed after the burning alive of the captured Jordanian pilot.
Although an immediate robust Jordanian military response is appropriate, it's not at all certain that the kingdom will keep up the tempo of operations after the fury over the pilot dissipates.
Over time, concerns about force preservation may ultimately compel the kingdom to dial back its own expanded military efforts in Syria. Jordan is unlikely to become a regional Sparta any time soon.
The writer is director of the Program on Arab Politics at the Washington Institute for Near East Policy.