You need to quit with the no lying/confess business or she will never tell you anything and it will be completely counterproductive.
It's wee on a towel,which is no big deal really, CLEARLY she doesn't want to tell you and perhaps, as by your own admission you have issues, this is precisely why she doesn't want to say anything as you will find it (her :() disgusting.
It was in the laundry basket so in the right place. Even a bit of wee would smell vile after a day or so in the basket, a wet flannel smells RANK and that's only water, so I doubt she actually stood over it and used it instead of the toilet.
The no lying thing is completely unrealistic and is ridiculous because we all lie whether out of kindness e.g. 'does my bum look big in this?' or ' oh fuck I've mopped up an accident, I'm so embarrassed I won't tell anyone'
No one tells the truth all the time and it's completely understandable that she didn't want to tell you.
Teenagers are pretty swanky, it IS like having a toddler around the house again as someone said upthread, all previous good behaviour goes out of the window and their brains are all over place. Honestly, if I had found a towel with wee on it in the laundry basket I would have bunged it in the machine and thought nothing more of it unless it kept happening.
Imagine if you had an accident OP and bunged your knickers into the machine to go through the next wash and your dh found them and made the fuss you are making, it would be embarrassing, wouldn't it?
Wrt personal hygiene - she needs to shower every day to get into a good routine, my ds is 12 and every day we have to tell him to go and have a shower,it just doesn't register yet that that is an essential part of the day - even though he needs 2 showers a day atm due to bad hay fever, it gets the pollen off him. It's not 'laziness' it's them not knowing how important personal hygiene is yet. So tell her to shower every single day without fail and you will be doing the sniff test
if she doesn't, tell her she can't have her phone/telly/whatever.
Sorry, really long post! Hope you feel calmer this morning about it all :)