Require every gun owner to register every firearm annually, undergo a background check to see if they will be allowed to keep them annually, pay a licensing fee annually, an insurance premium annually, and require paperwork for every gun transaction, whether sale, gift, or inheritance.
That's for existing guns.
Henceforth, every time a gun is found that does not meet the above requirements will be confiscated and the possessor shot to death on the spot, no further process needed. If the gun was registered to no one previously, that stops there. If the gun was previously registered as above and no transfer done and no theft report filed, execute them on the spot, too.
Every time a gun is used in a crime, even just showing it or just saying you have one, execute them on the spot, too.
I think in a relatively short time, you'd be surprised how law-abiding gun owners really can be, and anxious to be so.
Program is self-funded, registration and licensing and background checks should turn a tidy profit for the government. Prisons don't add a single inmate, so no increased cost there, either. All in all, a perfectly reasonable , modest solution to a problem.