If it is really impossible to tune this instrument take it to a luthier and let him check it. It is definitely possible to tune a tiple in some minutes and then play in tune for some hours.
Your instrument is an American Tiple. The tiple Colombiano has 12 strings, in 4-courses of three strings each, and is tuned C - E - A - D (traditional), or D - G - B - E (modern). It is also much larger than the American tiple, being nearly the size of a classical guitar. The American instrument is basically a ukulele variation, and traditional tuning was A - D - F# - B (which was also common on soprano ukes, up through the 1970s), although a lot of people tune them as you do, to modern uke tuning.
I've heard the joke before, but when you said, "You spend 15 minutes tuning it and 15 minutes playing out of tune," it me laugh so hard. It's a strange looking instrument. Great video!
