After replacing battery, clicking dosen't work even with mouse

Your trackpad is stuck in the down position so its always in click enable mode. When you replaced the battery you likely damaged the trackpad. Here’s a view of the bottom of the trackpad:

The oval is the blister switch which is the click.

You’ll need to remove the battery and carefully the metal plate to get to the trackpad out.

But first give this a try: Use a cotton swab which is just damp with water (on the dry side) you don’t want it to drip into the electronics below. Rub around the edge of the trackpad to see if its jammed up from junk. I also find a stiff piece of paper can push out any grains of grit as well.

To prove it you can disconnect the Trackpad MacBook Pro 15" Retina Display Early 2013 Logic Board Replacement Step26 At that point your external mouse should work (make sure you disconnect the battery first).