This is not easy to solve without actually having the bike to play with. It's also something I'd do fairly quickly so I'm trying to give a detailed explanation of something I've learned to do from experience rather than written instructions. I've built my own bikes, worked for a year as a mechanic, and ridden rather a lot on a variety of different bikes. I'm also regularly wrong and still learning. So...
Modern derailleur systems have a lot of things that have to be adjusted fairly accurately for them to work properly. Unfortunately some of the tests that a mechanic will do are destructive, in that they will usually cut the gear cable and replace it. Generally speaking for anything more complex than tweaking the barrel adjuster they will replace the cable and outer, because those are often the problem and they're cheap. In other words, paying $5 or $10 for a new cable+outer will often save $20 or more of mechanics time.
The other thing is that a cheap shifter like yours is only about $20 to replace, so often a shop will just put a new one on and not bother trying to fix it.
Leave the cover off the shifter (as it is in the photo), everything should still work. If you lost that cover in the crash, buy a new shifter (or replace the cover if you can somehow find a matching one). The missing cover isn't a problem in a nice clean shed, but on the road the shifter will fill up with water and muck and stop working fairly soon.
Any of these steps might fix the problem, but in order this is what I'd do:
Look inside the shifter, jiggle the bike, trying to see whether there are any loose parts in the shifter (or random debris). Note that it's full of sticky grease so anything loose will likely be stuck in place by the grease.
Try to work out what everything does, and whether anything is out of place.
If you find something, try to work out whether it broke off the shifter (and if so, whether it's important), or whether it's from the outside. Then remove it.
inspect the cable and outer between the shifter and rear derailleur. If there are kinks or damage that will probably be the problem, and it's the thing to fix first.
Next, get slack gear cable next to the shifter. Often there will be open sections of cable and you can pop a section of outer off the bike to get this. The attachment to the frame looks like this, although they're usually welded on rather than riveted:
Now you can pull the cable to make sure it slides freely in each section of outer. If it doesn't , replace it. What you're looking for here is not "maybe a little extra friction" but "hard or impossible to move by hand".
The shifter. Look for obvious damage. Hopefully you did that already :) Specifically, if the release lever that's not returning is bent it might rub on the body of the shifter. Even plastic levers do this.
If there's abrasion on the plastic at the join between lever and body that can also produce this effect, and carefully trimming away the damaged plastic along the join can sometimes fix it.
Click through gears. Without the derailleur pulling on the cable it will only shift into the lowest gear, but it should do that easily.
Pull gently on the cable as it leaves the shifter. You're just reproducing the pull from the derailleur spring, not ripping the shifter off the handlebars. Now click the other lever on that shifter to change back up through the gears.
It sounds as though the last step is likely to fail. If so all you can do is poke around in the shifter more aggressively, trying to dislodge anything loose, maybe put a bit more oil in there to looses the grease and see if that helps.
I'm sorry this is so non-specific, but in my experience the "generally poke it and lube it" approach works about half the time, and the rest of the time the problem is a damaged part that can't be replaced because no-one sells parts for cheap shifters. In the shop I worked at we had a box of old shifters for times like this and if the customer couldn't afford a new one we'd try to find a second hand one. But that saves $10 on a bill that's already got over $50 in labour costs on it.
