So I tried few things about the persuading minigame.
And here is the procedure which work for me
1. Buy the persuasion power from Veronica
2. Play the persuasion minigame until you obtain enough trust points
3. Then choose Hire
If you choose persuade the girl without persuasive power it will close the dialog without any notification why.
I'd like to add that persuasion points are expendable amount of tries in card minigame. Yes, you expend them after you buy them. Its not a stat as someone might think at the first glance, it is a resource, basically.
It does not sound too good, since they cost 250 gold each, which turns into quite a sum when you buy them a lot. And trust points you earn through the minigame are also an expendable resource.
But there is the kicker: both types of points are not only expandable -
they also stay with you.
For example, when I started brothel quests I used a shitload of money to buy some 40-50 persuasiot points. When I recruited dwarf girl, I spent most of them on acquaring, IIRC, 17 trust points.
It means that after I hired dwarf girl for 3 trust points, I have another 14 left to hire several more girls later.
TL;DR: just buy a lot of persuasion and try to win a lot of trust on your first quest and do not bother about persuasion for a while.
Just wanted to make matter clear to whoever is going to read this post.