I think you ultimately need to decide if you like your job and want to stay and work there vs. taking on a new opportunity elsewhere. If you want to stay but the salary is only the problem, then you absolutely ask for a raise. Do you like your job? Is the main complaint the salary?
Assuming you decide you want to stay and you go for the raise, I would:
1. Schedule a formal meeting to chat
2. Let them know that you've been doing some research and learned you're underpaid for the job. Don't give them specific numbers like $10k. That boxes you in. Just say that you're underpaid for your level AND that you outperform those at your current level so you you're looking for your compensation to better match your value. i.e., you want to be at the highest end of competitive for your level.
3. Have all the data ready, but don't present it unless you need it. Otherwise, I think it's a little awkward. You don't have to convince them of your value--they should already know it. However, if they ask about your research, then verbally tell them the things that backup your case. Then say that you'd be happy to provide documentation if they want it. Don't give them data that has specific numbers on salary. Again, that boxes you in. If someone at a lower level makes more than you, they know you're underpaid too. Remember that.
4. Stress that you like the job, the work, people, etc, but that this is something you'd like to see addressed. Just leave it at that.
5. Finally, DO NOT GIVE THEM A NUMBER. Confidently present yourself, let them know you want more $$ (or other types of compensation if that's available), but DO NOT give the first number. Let them come to you first and then counter higher. Otherwise you risk lowballing yourself. Once they come with a number, you MUST counter, even if it's higher than you thought.
As far as being a woman, research shows that women can be successful negotiators. One thing that women often have to do more than men is to provide strong reasons for the raise, and you've got that covered. So if you decide to go this route, just know that the market is on your side. In my experience, the worst thing that can happen is they say no and you respond by aggressively looking for another job.
Btw, you're right to focus on your value to the company vs. what you need or want in your personal life. I can't even imagine doing the latter. I'd feel like an idiot.