You're reading this right, the battery was reversed...

To start from the beginning, our Accord had been sitting for quite awhile, and the battery was in need of replacing. I told my daughter to run it to the local advanced auto parts and let it charge overnight. My special snowflake picked up the wrong battery that belonged to a toyota and put it in the car. Based off what she said, the car started and drove back home, but on the dash there were several lights on. After realizing that the battery was wrong, she took it back, got the correct one, and again, the car drove home perfectly fine, yet several lights on the dash were still on. After finding out the original battery had a bad cell, we got a brand new one. She brought the new one home and I installed it. Now I know I put the battery back on its correct terminals, but I'm not sure if my daughter put them back correctly when she put that toyota one in...

I started up the car now at this point and let it run for a minute, and then I proceeded to try to turn the key off to the off position, and that did not turn off the car. I went over to the battery terminals and disconnected them, and the car died after 30 seconds.
Currently the car cranks, but won't start. I turned on the A/C, and it blew a puff of white smoke. I checked out the ECU box, and it smells burnt. The dash won't show the mileage, and the green key symbol won't come up on the dash.

What steps should I take to get it up and running again. I'm not quite sure where to start with this. Obviously its an electrical problem.

 

If the battery cables were reversed, the large battery fuse (under-hood fuse no. 41 (100A) ) would have been blown. Since the car still have power and that fuse was not replaced, there is no evidence of the battery cables being reversed.

Check passenger side under-dash fusebox, Fuse No. 13 (7.5A) - back-up fuse to see if it is blown. There are two separate fuse boxes inside the car (driver's side and passenger side). It is possible other fuses are blown.

 

Thank you for the suggestion. I will try this out and reply back with results.