I don't quite understand how Hezekiah relates chronologically to Shalmaneser.
In Kings Chapter 18, it says that Sennacherib attacked Judah in the 14th year of Hezekiah. So, if this occurred in 701 BC, then Hezekiah would have become king in approximately 715 BC. In the same chapter it says that Shalmaneser attacked Israel in the 4th year of Hezekiah and captured Samaria 3 years later, so those events would have been in approximately 711 BC and 708 BC, respectively.
Thus, that would mean that Shalmaneser was king of Assyria at least until 708 BC. Since in Sennacherib's Prism, it says that his campaign against Israel and Judah was his third. Thus, there can be at most 4 years between Shalmaneser and Sennacherib.
However, between Shalmaneser and Sennacherib ruled Sargon, mentioned in Isaiah, and Sargon is well attested to have ruled approximately 16 years. So, how can we have a reign of Sargon lasting 16 years, when according to Kings, there appear to be only 4 years separating Shalmaneser (who preceded Sargon) and Sennacherib, Sargon's son who was his successor?