I've tried this problem 3 times before looking at how the professor solved part (a), and every time I keep coming up with the same KVL equation. Why is my KVL wrong?
Your professor used a loop from the base of Q1 (which is given as 0 volts or ground) back to ground through the -5V supply. For this case the appropriate Q1 voltage is Vbe. The advantage of using this loop is that only one unknown (Ic1) appears. You chose to use a loop from +5V to -5V. In this case, however, the appropriate Q1 voltage is Vce1, not Vbe. That is your error. Also, it is not as useful since this loop contains 2 unknowns (Ic1 and Vce1).