Now that he is thoroughly wet, let him sit in the shade or light sun. If it doesn't have functioning leaves, it doesn't really need sunlight. Wait to water again until the soil is dry on the top, but just a little moist underneath. You can use a dry wooden chopstick to test the soil. Leave it in for a while, then pull it out and it will tell you whether the soil is wet underneath the surface. IF the leaves are too wilted, misting them will not help. If they just look like they could perk up, they will show improvements by tomorrow. Don't remove them unless you are sure that they are dead. Remember, brown leaves don't hurt the tree, otherwise beeches would be crying all winter long.
You have time, to wait and see whether they will perk up or whether the tree will have to grow new buds instead.
You might want to consider a growing light if you absolutely must grow the elm indoors. The summer sun is going to be difficult to regulate... a few hours of sunlight reflected through a window will cause a lot of problems. So, consider using a growing light instead of the windowsill. You can search this forum for 'growing lights' or 'indoors' to find information.
Good luck!
Joanie