A professor had agreed to an internship and asked me to remind her again before the start of August. She has not responded to the reminder email I sent 10 days ago. Should I mail her again? Is it impolite to call her in her office? I was very eagerly looking forward to work with her, so yes, I’m slightly desperate.
It is very likely that she just overlooked the mail. You definitely should send her a mail again. It's totally reasonable to keep asking until you get the response, even if she suddenly changed her mind about hiring you.
BTW, it's summer now. Many professors would go on vacation at this moment. If this is the case, I am afraid you have to wait until she gets back to work.
Sorry that this comes so late, but others may benefit.
In such situations it is advisable to contact the professor's department, usually via a public phone number or email address. A staff member in the department might be able to put you in contact if you explain the situation. Or, they might be able to explain to you why communication is impossible at the moment.
Don't make it a complaint against the professor, just an inquiry about how best to communicate. The staff member might have a lot of options for such things, including contacting the professor on your behalf or asking the department head for advice.
But this is for time-sensitive inquiries, such as the one in the question here. If waiting is reasonable, then wait until it is closer to deadlines that can't be missed.