How should I write the last sentence in my email to a professor?
As user11192 says, the closing sentence does not matter much.  I'd advise you to focus on proofreading your entire letter.  Not to be unkind, but there are typographical mistakes in two of your three closings:


  I hope this helps. If you need further information, Please feel free
  to contact me at: My email


The first letter in "Please" should not be capitalized.


  I hope this helps you evaluate my eligibility for financial aid better.


This is fine, except, as aeismail points out, financial aid is need-based, while you are applying for a job, which is presumably merit-based.


  I hope this helps, please let me know if you need more information.


This is a comma splice.

In a closing, I would also thank the professor for his/her time:


  Please let me know if any further information would be helpful.  Thank you for your time.