"This is very interesting as a project" What is the next step to obtain PhD position?

Question

I am submitting to several schools for PhD position in CS. I am following the advices about how to contact professors.

One professor I sent to suggesting some idea about her previous research and asking that I want to indicate my desire to work under her supervision. She replied "This is very interesting as a project. Please apply for the next deadline". It is very encouraging response. However, the admission decision is at the department level. So I will have to apply and there are several other requirements which I am not completely sure if I entirely meet.

I feel like I must focus on this positive side in order to increase my chances, but I do not know how. Should I keep in touch with the professor although she can do nothing now until the decision is made? Should I send more details about the suggested idea in order to show how motivated I am (I am not sure if she's part of the admission committee). What is the second step to increase my chances to get admission?

Do you think sending more emails to the professor is annoying or does show my enthusiasm? Any thing I can do to support more application at this time?

Update for who may concern
The application has not been accepted.

Answer 1

The second step is to submit a strong application: Good grades from a good department, a compelling statement of purpose (which reiterates your interest in your target advisor), and strong letters of recommendation that address your research potential in personal, specific, technical, and credible detail. And then write the professor to tell her you've applied and reiterate your interest.

As long as you're building a strong application, you should send it to several other schools as well (minus your interest in that particular advisor, of course). There is absolutely nothing you can do to guarantee admission to your first-choice program, so you need to cultivate other options as well.