In general, it is more polite to ask for help achieving "something" than to ask directly for the "something".
I would like to ask you if it is possible to secure me an internship with a software company.
could be rephrased as
Would you be able to help me secure an internship with a software company?
First, we turn our thought into a question so that it seems less like a demand, and then we ask if it is possible that she might be able to help. This way even if she can't directly find you a position, she may still be able to help by introducing you to someone who can.
I would also rephrase the second and third sentences:
I will have approximately two weeks free between first and second semester, and I would like to use that time to increase my experience.
This isn't English advice, but I can't help but add that in my experience in the US, two weeks is a very short time for an internship. Instead of asking for help getting an internship, you may want to ask for her advice on how to get practical experience during that time.
I was a part time intern for a professor in the Engineering school while I was pursuing my degree, and it worked out well because my work was close to my classes and the university had a process in place for hiring students. A position with an outside company is more difficult to secure because there is a lot more cost in hiring a new intern and they will likely want a longer commitment.