Why did my chocolate tempering fail? I just tried today to temper dark chocolate for the second time. I used about a pound of Toll House dark chocolate chips. If the quality of that chocolate is my problem, that would help me a lot.

My first attempt, I heated 3/4 of the chocolate to about 115 F, took it off heat, and tried to cool it by adding the remaining chocolate as a seed. The instructions I used said it should cool and almost melt all the seed. Wrong, it all melted in about 10 seconds. I'm guessing all the temper in that seed was lost and my chances were ruined right then. All my test parchment failed to set up. I also noticed the thermometer I was using was awful so I went and bought a fancy infrared gun thermometer. I put the failed bowl in the fridge to set up.

I just tried again with the new thermometer. Having no tempered seed, I figured I'd try the melt, cool, reheat method. I put the chunks of chocolate into the glass bowl over a pot of a water on very low heat, and melted it up to 115 again. Then I took it off the heat, and spent over 20 minutes stirring and waiting for it to cool it down to about 82.5 degrees. Then I put it back on the heat for 10 second increments, heating it first to about 85, and keeping it there for a few minutes, then heating it to just a fraction over 88. I held it at that temperature for a few minutes to make sure the form IV crystals were gone, and then dipped some parchment strips. Again, they never set, even in the fridge. Like before, they looked ok, with good gloss, but stayed soft and smeared to the touch.

Rather than getting discouraged I tried heating it back up to about 90 degrees, held it again, and then did more tests. Still nothing. It's back in the fridge now.

I have no idea what to do next except get a different batch of higher quality chocolate to try.
I'm going to blame your thermometers.
The one you used for your first batch you believe was a bad one, so it likely was measuring the temperature incorrectly, so you got poor results.
Then, you bought a really fancy infrared thermometer... which is great for measuring surface temperature but is pretty useless for anything else. Note that surface temp is usually much different than internal temp. Generally, if you're heating something up, the surface will be hotter than the inside and if you're cooling something down, the surface will be cooler than the inside.
This means that all your temperature readings were wrong!
Here's some info about the internal temperature reading myth:

2. An infrared thermometer will tell you the internal temperature
This is another myth worth busting. An infrared thermometer is a surface temperature tool – period. If you’re grilling, baking, smoking, or roasting you’re going to need a penetration probe to tell you the internal temperature of the food you’re cooking. An infrared will only give you the surface temperature of the food, and depending on your optical range, the temp of the surrounding grill, skillet, oven, etc.

I think you should try your chocolate tempering again but with a good-quality instant read thermometer (make sure the temperature range goes low enough) or a candy thermometer (again, some of them start at 100 F, so make sure it goes to the temps you need).