but if you want to write a likable jerk character give them punishments for their actions
Or
don't give them punishments for their actions. Not in the Aesop sense at least.
In their natural environment, bratty people usually go without real punishments beyond minor social hits that either do nothing to fix their behavior or ends up reinforcing it for a myriad of reasons. Hell, they're often rewarded, quickly discovering that life is a social game that you can cheat at. If they have a heritage/position advantage, have them milk it for all its worth. Make it their coping method for neglect, a "strategy" imparted upon them by a mentor, or a result of fear/mistrust/vindication for once being on the bottom. This isn't a bad thing as long as you know how to play into it.
Whether or not a brat "pays" for their actions should depend on if they get caught out in a situation where they can be held accountable or if they suddenly lose what let them get away with their crap. For extreme examples, have your exiled prince demand the respect of a mercenary that's already independently wealthy and beyond his legal jurisdiction or place your isekai'd CEO's daughter in a Xianxia sect and turn the plot armor off. Just be aware that punishing bratty characters might satisfy an audience, but it won't be enough to make them like the character, not on it's own.
On the opposite side of things, if you create situations where their behavior
would logically warrant immediate retribution but make everything go their way regardless. Your jerk/brat/douchecannoe will usually fail as a character (as seen in
Goodbye Deponia and
YIIK).
Bratty characters aren't (only) fun because they're a natural source of conflict, they often come with interesting baggage to unpack. Try to spend some time on why she's the way she is and have that flavor her interactions with other characters. Maybe the kid was taught to look down on others, maybe the kid is used to getting her way all the time, maybe her parents are dead and she's been put into a position of power and she's tired and scared and lonely and her prickliness is an attempt at acting in a manner that she
thinks befits her station.
If you need to boost her likability right away, sprinkle in some obvious kindness/genuine desire to connect but place some irrational problems in the way. Have whoever has to take care of her balance their empathy with irritation at how hard it is to get her to act like a responsible person- just make sure you don't generate melodrama by underdevolping a social problem or treating it like a one-and-done schtick, never to be referenced again after it's resolved.
