Another Illionoisian! Welcome! The HRM215 and GXV140 engine is a great combination.
I'm sure you've checked to make sure you've got good spark, and that gas is actually filling the carb bowl. Honda engines do like to have chokes that close all the way when they are cold. If you had the carb off, and throttle cable disconnected, make sure the throttle cable is adjusted to get the choke to close all the way.
You can poke around on this site for diagrams of your carb, and a nifty video showing the cleaning proceedure.
Have you tried a shot of starter fluid? Wonder what it would do wiith that? If they act like they want to run as long as the starter fluid is present, that usually tells me I missed something in the cleaning.
Two key things to a successful carb overhaul:
• It must be totally cleaned, especially the main jet and nozzle area. *Any* tiny bit of slime, debris, etc. can cause a problem and the engine won't start.
• It must go back together exactly right, and all gaskets must seal up tight. One screw not set right, one part not secure, or a tiny gap or tear in a gasket will also prevent the engine from not starting.
Sometimes, it's faster-cheaper-easier to simply replace the carb and install new gaskets. Depending on the engine, a new carb and gasket set can run less than $30, and if you spend 1-3 hours trying to get an old carb working again, it may be wiser to just replace it.
If you will provide the serial number off the back of the mower deck, I can tell you the part numbers and list prices for a new carb and gasket set. Include your ZIP code and I can list out dealers close to you who may have the parts in stock.
As others have said, be sure to verify the choke is working properly. It must fully close when the engine is cold, and many times even when hot, for the engine to start easily.
Do you have spark? There is a switch that operates mechanically that shuts the ignition off. Depending on which model of HRM215 you have, it is either operated by the throttle cable or the flywheel brake system. The lever that operates the switch is attached to the larger operating mechanism via a light weight spring that may come off if you've done any fidling with those parts of the mower.
If that's working as it should, and you have no spark, test the coil. I had one that started fine one day, ran for a bit, then it was as if I had turned it off and the coil tested bad.
I know it's an old thread, but perhaps this info will help someone searching for an answer.
I had the same starting problems with my Honda Harmony 1011. At one point years ago, it turned out to be the gas cap, which wasn't venting properly and air couldn't get in. New cap fixed it, and now sometimes I leave the cap a little loose just in case the cap gums up again.
More serious problems in 2018 with starting the engine. I could start it when the mower was cool, and it would run for 15 minutes or so, then would die and not start again. Sometimes I could let it sit a few minutes, and it would start; other times this didn't work. This happened all summer. This spring, I replaced the coil. It was amazingly easy to do and even the actual Honda part was not very expensive (about $40 I think). One of the threads I read said to use a business card to gap the coil, which I did. I have not had any problems with the mower this year. So, it was the coil.
Probably could use to overhaul the carb as others suggest, but for now it's working and I don't want to fix what's not broken!