I love Jordan and am considering it for a long weekend in autumn. I went with my mum years ago but have held off taking the kids because I think they really do need to be older for them to appreciate it. However, in my experience a 2 year old rarely appreciates anywhere you take them so you might as well go where you want to go if you think your toddler will manage it. I would say that the Jordanians are lovely people and the Bedouin are very welcoming so your toddler would be well spoilt by the locals. However, here are my thoughts on Jordan with a 2yo.
Firstly, I remember a lot of driving to fit in all the sites. Jordan is a small country and there is much temptation to try and fit everything in, but a 'short drive' can sometimes mean 4 hours of driving across fairly featureless desert.
Petra is amazing, but it is a kilometre walk from the entrance of the Siq to the Treasury (the first building you come across.) I would say in order to see most of the sights here you would be walking about 6 or 7 miles in a day, and climbing some hefty mountain paths. We went at a cooler time of year but it was still very hot and quite strenuous. A 2 year old would not walk round Petra, so you'd miss out seeing the full site.
Wadi Rum is fabulous too, but it's more driving, and unless your toddler is big on deserts, Lawrence of Arabia, and rock formations its likely to be another boring day. At the Bedouin camp on the evening s/he would be treated like royalty though.
Aqaba is really only a great destination for the keen scuba diver. The beaches are rocky and steep, you gain entry to the sea to snorkel via floating pontoons which have lion fish (poisonous) lurking under them.
The Dead Sea is great fun and I can see how a toddler would have a good time here, but the salt stings so make sure s/he doesnt have any cuts or gets the water in their eyes.
Karak is impressive but a long drive from anywhere.
Jerash is also impressive and is much closer to Amman, so you could stay in a resort style hotel and still get out there for the day, same with Mount Nebo, Madaba, Bethany-beyond-the-jordan, Al-Joun and various other places less well heard of.
I am certain you'd enjoy the country, but it could be quite testing with a 2 year old in tow. I can think of lots of other cultural places that have much more toddler-friendly activities - Mexico, Greece or Italy, for example. But it's up to you.