I would second the T4i with the 18-135 STP.
If you want Nikon, I would look at the new 3200 as I believe that has auto focus in video mode.
I know there are those with the opinion that manual focus is better, and that may be for the experienced, but it doesn't sound like that is you at this point (it's not me either
I have a T3i which does not auto focus in video. I was using a 75-300 lens and shooting video of my 6 yr old running toward me from 3rd base to home plate in her T Ball game. Now, I don't know about the other cameras, but the T3i makes you use live view when shooting video (I think by design that would be true of all DSLR cameras). On a bright sunny day, you cannot very well see just how in focus things are and that is enough of a distance that the subject will go out of focus. Plus, until you get experience, I keep turning the focus ring the wrong way until I figure out what is right, all the while it is recording, and still it is difficult to see. Love em or hate em, my Panasonic GF3 is really better for this as it will auto focus, but I don't currently have a lens longer than 42mm.
Everyone is different, but my guess is you are going to want video of your child. Maybe you are better off with a dedicated video camera, but remember that is more to carry. The beauty of the 18-135 is it covers a pretty decent range and with the T4i you will probably get decent video results. They may not be cinema quality, but then what are your expectations? Aside from maintaining focus (a work in process) I have been very happy with the video from the T3i when viewed on a 46 inch 1080P TV.
You may want to consider waiting for the replacement for the D5100 as it is probably not far of.
Being the owner of a 60D, and happy at that, oddly I'd recommend the T4i w/ the new 18-135 STM lens (though I understand why you might want the 15-85 for better IQ). This is based on several things:
The 60D for video is fine, in my opinion, for fixed focus situations. But when you try to follow a toddler/young child around the lack of AF is going to frustrate. Of course we have no reviews of how the hybrid AF in video is going to work on the T4i, I'd hope it's at least decent and very usable for your situation.
The T4i frame rate is now pretty good making it easier to capture the quick action of young children.
Just some thoughts. I have no opinion/experience with the other two options.
Regarding the D5100, I thought that was the direct competition for the T3i now T4i. They are priced pretty close. I thought the D3100/D3200s were to compete with the T3. The 60D I thought was to compete with the Nikon above the D5100 (7000?).