Targeted drug therapies have proven very useful as cancer treatments, but some still demonstrate very little antitumor activity in certain cancers. A class of tyrosine kinase inhibitors (TKIs) that are unexpectedly inefficient in some cancers are those that target the human epidermal growth factor receptor (HER) family. This paper proposes a mathematical model of the HER signaling pathway for the study and treatment of cancers that overexpress. The goals of this paper are threefold. The first is to introduce an ODE model that agrees with the existing data over large time intervals, and the second is to use this model to understand the effects of applying certain drugs to such cancers. We finally use this model to determine a "steering-based" treatment scheme for cancers that overexpress HER2.