We give a degree formula for Grothendieck polynomials indexed by vexillary permutations. We apply this formula to compute the Castelnuovo-Mumford regularity for certain classes of generalized determinantal ideals. In particular, we give a combinatorial formula for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of certain Kazhdan-Lusztig ideals, including those coming from open patches of Grassmannians. This provides a correction to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri (2015). This is joint work with Jenna Rajchgot and Anna Weigandt.