Forgive me for offering you an answer which, one might maintain, is as unresponsive as some of our yo-yo’s, but string lasts… as long as it lasts.
It simply depends too much on factors related to how you throw, your particular style of play, perhaps even on local environmental conditions, and a host of other things, to permit a meaningful estimate of a string’s life expectancy. To say nothing of the fact that what, to some people, constitutes a worn out string in immediate need of replacement, will to other people seem like a string which still has plenty of life left in it.
Examine your string, and decide - based on your experience - whether it seems to be in need of replacement or not. That’s really the only way to handle the matter.
I would add that many beginning guitarists ask much the same question - when should I replace my strings? They’re often unwilling to accept pretty much the same answer as I’ve given you - replace them when they look, sound, or feel like they need to be replaced.
It’s never occurred to me to wash my yo-yo strings, any more than I would think to wash and re-use most other extremely inexpensive items intended to be used until dirty, then thrown out, like a paper plate or dental floss. I regard string as a disposable. What does Fat Kitty string cost, in quantity of 100 pieces? 20-25 cents, maybe? Even in small quantities (10 strings), I doubt whether it’s much more than 30-35 cents per string. Just not enough money at stake to motivate me to do other than throw a string away once it’s sufficiently soiled to merit washing.
I mean, I’m not trying to be like the cartoon character who disposes of his Rolls Royce because its ash tray is full. Although if you happen to know of a deal where I can buy Rolls Royces for 20 cents each…
Or am I missing the point? Is washing the string supposed to enhance its qualities or something? (I’d find this difficult to believe, but I’ve heard stranger claims.)