Get a coat hanger, and wrap it in a single layer of electrical tape, so it doesn't scratch the bore.
Typically a .22 shell is very easy to pop out and doesn't expand much after firing. With the proper sized flat head screwdriver you should be able to get into the extractor groove in the barrel, and under the rim of the cartridge to get it out.
I guess you never cleaned the gun before shooting it?
How many rounds did you get through?
I imagine it is the packing oil, collecting all of the gunk from some cheap .22lr that eventually built up in the chamber, that caused your issue.
Lesson #1 - always clean a brand new gun, especially the chamber, and barrel.
Lesson #2 - remember lesson #1