I don't know if your headset is "Aheadset" brand or not (I don't see any markings); you'd have to look at the various Aheadset brand headset images online to see if any match.
But you should be able to use any brand headset that is designed for your headtube and steerer. A handy cribsheet of sizes for headsets at this page
Sheldon Brown's Bicycle Headset Dimension Crib Sheet
that may be useful.
What size was your old threaded steerer? (it's usually stamped into the steerer tube somewhere, otherwise you can measure it's outside diameter (OD).
What is your new steerer's marked OD, if any? (otherwise you'll have to use the measured OD, if that's 28.5mm, which is typical for threadless 1-1/8" steerers)
Some steerers have a larger diameter at the base than the top, so your headset (and headtube) have to be made for that.
If you're going to replace the races already in your headtube, then to know what headset to get for your headtube, you'll need to know the ID of that. It's probably very close to the ID of the inside of the top lip of the upper race, if you can't measure the actual headtube ID. One of your measurements might be measuring this, but is blurry so I can't tell for sure; if it is then its about 31mm.
You also have to know the size of the very bottom of the steerer, so you get a headset with a matching crown race for that.
Depending on what parts you have already with the new threadless fork, you'll need more than just a headset to install it.
Does your new fork have the star nut installed, or do you have one to put in? (and the top cap and tensioning bolt)
Do you have spacers to fill the gap between the cap and the stem, and/or stem and headset?
Also, your handlebar stem from the old threaded one is probably a quill type that goes inside the steerer tube, and won't work with the new steerer, so you'll need a new stem that clamps on the outsde of the new steerer. The stem has to have a clamp for the bars that's the right diameter for the large part in the center of them with the scribed lines on it; this may have a stamped marking with it's OD to help.
Below is a diagram found in an image search for "Threadless steerer" to show you the full stack of parts you need.
Another image from Sheld Brown's bicycle maintenance site
types of bicycle headsets (steering bearings; how to rebuild and adjust them
www.sheldonbrown.com