End mills are expected to be delivered Fri. the 4th.
Quill Stop - Could tell you did not use quill stop as the .062 drill was broken. 99% of small drills broken are by hand feeding by feel without a stop. So knowing it was not used was easy, but not knowing the machine does not have one is a surprise. Would like you very much to have success in removing the small drill which is about 1/4 inch long. As a post saying " Broken .062 Drill Bit with Broken Carbide Ball Mill Removal" is not the help you need.
Would feel very confident using 1/16 mill with the following set up.
The End mill can be chucked up and run perfectly true.
The machine runs smooth at the 3,000 rpm with no radial slop in head.
The setup is true and solid. Many times a 5 minute job can take an hour to set up. Which is why in 5 minutes it is done.
And most important a solid stop, that can be relied on. Since you have suggested on improvising one it needs to be a solid quill stop that can be relied on doing the job. If you were to use a 3/8 fine thread bolt for adjustment in a solid fixture that would be good. As 24 threads per inch turning a head flat at a time will advance .007 inch at a time. Remember the .062 drill that you will be removing is 2 flutes with all kinds of grab spots and will be requiring you to sneak up on a touch. After that you need not worry about feel, the stop is all that is needed. Have done this many times for people as a 35 year retired tool maker and yes after setup it is a 5 minute easy job. If you follow the above you should have 99% success and yes I am on your side. If you can not adhere to the above than stop. Go down the mountain where a proper machine can be had and used. Since you do seem to have knowledge of your previous life, please check to see if you are a person that often makes the same mistake twice. And if it breaks and cannot be pick out, it's not the end of the world just look up EDM Small Hole Drilling on the internet.