One week based at Dead Sea

Could work.
You can combine Jerash with Ajlun. As it is just one day you mightwant to do this by taxi.
For Petra I suggest you rent a car and have it delivered to the hotel. Take the Dead Sea road south and get to Petra via Wadi Araba. Should take you a relaxed 3 hours.
On Saturday drive back via the King's Highway. Check bing maps for your planning.

Petra By Night was.... meh. It's kind of cool, and quibbling over 12 JD seems silly - if you're there that night, you might as well go. Jerash and Ajloun in a day is no problem at all, they're relatively close together. If you leave reasonably early you can also make a trip to the Amman Citadel which is well worth it - the museum and site closes at 6pm, so plan accordingly.

Petra you don't need to be up at the crack of dawn for - the best time to see the Treasury is around 9:30-11am anyhow when it's full sun (though I thought it was more stunning in the afternoon, personally). Make the trip to the Monastery or rent a donkey for 10 JD - and if you've still got it in you - go to the High Place Of Sacrifice as well. You might still have enough energy for the Royal Tombs - but that's my suggestions in order.

Bring lots of water to Petra - it's a tremendous ripoff once you are inside for obvious reasons (about 4x the price - 2 JD for a 1.5L bottle that you can usually get for a half dinar elsewhere.

Madaba can be easily combined with Mount Nebo. Also, if you want a Dead Sea mud pack without paying hotel spa prices - the do it yourself version is available at Amman Beach - 16 JD to get in and 3 JD for all the mud you want.

If you need a driver, call Jamal Abu Hamzah 00962 796008501 or 788385367 (you might need to add a 0 first, if I remember right). His email is jamal_salamah@yahoo.com - his rates are reasonable and he's a really nice guy. His English isn't the best, but definitely reasonable. I had him for a four day whirlwind tour and he was top notch.

If you are based at the Dead Sea the Desert Highway is certainly not the fastest way to get to Petra.

#3 "Bring lots of water to Petra - it's a tremendous ripoff once you are inside for obvious reasons (about 4x the price - 2 JD for a 1.5L bottle that you can usually get for a half dinar elsewhere"
Do you really think that carrying a heavy load of water around is worth the trouble, just to save a few J.D. ? - especially considering all the steps you will supposedly be climbing??

Henkbos!
You might be right about Desert Highway I have only a feeling that it’s the fastest way.
However it´s only 26 km back to Madaba and another 15 km to Desert Highway and it´s a highway with high speed almost all the way down to Petra. The road along Dead Sea is fast but climbing the serpentine road up to Karak is slow after that you have to drive through Karak as well to take Desert Highway to Petra this distance is 60 km compared to 41 km for the Madaba route.
King Highway from Karak to Petra will be far slower than desert Highway since one has to go through Tafilah which is really slow and takes a very long time.
Best Regards from
swebackpacker

Yeah. It's worth it. I took three litres in a Camelbak, no problem.

@SWE: Not if you drive down to Wadi Araba and take the Namla Road, straight down to Petra :-)