If the problem is nothing more than having to double strike the cartridge with the hammer to get the pistol to discharge the .22 cartridge, that is a really easy fix for any competent welder or gunsmith: you merely add metal to the hammer and then file or grind it down to the original shape--the idea is to lengthen the hammer so it strikes the cartridge with more force, as it originally did--and the fellow who stated "his gunsmith FIXED his so that now it requires 4 strikes to fire the gun; well, that gunsmith was NOT competent. Another option is, if the hammer is long enough, is to replace the hammer spring beneath the handles with stronger spring metal. Again, just get the same gauge metal and shape it to the original, as the original may have lost some of its "spring" due to age. One option or the other will definitely fix the gun, depending on whether the spring has weakened or the hammer has been beaten down until it is not sufficiently long enough to strike the cartridge. BOTH are very easy fixes.
BigBob
The Butler 22 caliber single shot pistol was made by Derringer. The Derringer company was founded by Henry Derringer. Derringer pistols are the smallest handguns available and were most commonly used by women.
Check the on line auction sites for what they have sold for. Blue Book of Gun Values might have it listed.
The maker of your Derringer is no longer in business. Sorry-
10-50 usd
I have the same firearm I belive, I have fired Remington .22 short rounds. Mine has a slight problem with having to strike twice to set the round off He Actually Hates It! I also have one and it also has to strike twice to fire. Or it did. A gunsmith "improved" it to where it now requires 4 strikes to fire. You can get some parts as it is a copy of a Colt. Google "butler derringer parts".
50-100 or so
contact: cobra.pistols.net They have the parts in stock
20-70 USD or so
25-100 usd
$250.00 if in excellent condition
You need a gunsmith
About $250-300, depending on condition.