So, our sources are pretty limited for this period as a whole. They are:

1. The Hebrew Bible. Primarily the Book of Kings and the Book of Chronicles, plus mentions in some of the Prophets. Scholars generally agree that the Book of Kings was put in its final form before the Book of Chronicles, and therefore it was recorded closer to the events that happen and therefore is more reliable as a historical document. So when Kings and Chronicles disagree, people generally go with Kings.

2. Archeology. Though a lot of the interesting things, like "how people reacted", might not show up in the archeological record.

3. Written records of other neighboring states. These records to have only short mentions of what's going on Israel/Judah the are is a little backwater between Egypt and Mesopotamia.

So, where to start. I'm going to assume you don't mean Zedekiah, who was the last king of Southern Kingdom of Judah at the time of the Babylonian exile. He ruled to 589/7 B.C.E. and was put on the throne by the Babylonians, who had already conquered Jerusalem in 597. So Zedekiah-->Babylonians-->conquest of Judah, vs. Hoshea in Israel/Ahaz in Judah-->Assyrians-->destruction of Israel. I'm assuming you're asking about events related to the destruction of the northern kingdom of Israel which was then ruled by Hoshea, nearly a century earlier. Israel (also called "Samaria" after its capital city) fell in 722 B.C.E. The best place to start this history is probably in 745, when Tilgath-pilester III took the throne of Assyria. The last Assyrian campaign in this region had been 796, but in 743 Tilgath-pilester started his first western campaign. At the end of the campaign in 738, Urartu and Arpad (two of the most important players in the northern Levant) are defeated, we have a tribute list that includes Rezin (Ra-hi-a-nu) of Damascus, Zabibe of Arabia, and Menahem (Me-ni-hi-im-me) of Israel--but no one from Judah, where Jotham was king at this point.

2 Kings 15:35 only mentions that Jotham "built the upper gate of the house of the Lord" but Chronicles 27:3-4 says:

3 He built the upper gate of the house of the Lord, and did extensive building on the wall of Ophel. 4 Moreover he built cities in the hill country of Judah, and forts and towers on the wooded hills.''

The Ophel is basically the Hebrew equivalent of acropolis. This is often taken as a sign that Jotham is securing his defense in light of a possible invasion... but an invasion by Israel and Damascus not Assyria. And this is not, Pekah of Israel (Pekah took the throne from Pekahniah, Menahem's son) and Rezin of Damascas do start attacking Judah (2 Kings 15:37), probably to try to make him join an anti-Assyrian coalition.

2 Kings 16:5-9 records that Ahaz, now king of Judah, is losing land and pays tribute to Assyria to come his aid: instead of joining Damascus and Israel's campaign against Assyria, Judah voluntarily becomes an Assyrian vassal. Other sources tell us that Judah was a hold to a larger anti-Assyrian movement including not just Israel and Damscus, but also Tyre, Askelon and Gaza. Tilgath-pilester's second western campaign (734-732) that begins with the conquest of the entire coastal plain and ends in the fall of Damascus (long the strongest player in the region). In the campaign, we have records that he accepts tribute from Tyre, Ashkelon, Gaza, Edom, Moab, Bit-Ammon, and Ahaz (Ia-a-hu-zi) of Judah in 734 before turning his attention to the north and focusing on Israel and Damscus in 733-2. Already in this campaign, we have Assyria conquering and "carrying off" parts of Israel. This led to a coup in Israel (2 Kings 15:30) and Hoshea replaces Pekah as king of Israel (as a note to those not familiar with the region: after the United Monarchy, Judah is ruled entirely by one dynasty, David's, until around the Babylonian Captivity, whereas dynasties in Israel rarely are able to string three kings together before there's another coup). These conquered regions of Israel were incorporated into the new Assyrian province of Dor and Hoshea, with Rezin executed and Damascus conquered, became a vassal of Assyria. By end of the campaign, Tilgath-Pilester controlled all the outlets for the lucrative trade routes in the region, though parts of the routes remained under his vassals.

We have foreign worse records for Tiglath-pileser's successor, Shalmaneser V (727-722), and account in Kings (2 Kings 17:1-6) is brief but followed by a long description of their sins against God (2 Kings 17:7-18):

17:1 In the twelfth year of King Ahaz of Judah, Hoshea son of Elah began to reign in Samaria over Israel; he reigned nine years. 2 He did what was evil in the sight of the Lord, yet not like the kings of Israel who were before him. 3 King Shalmaneser of Assyria came up against him; Hoshea became his vassal, and paid him tribute. 4 But the king of Assyria found treachery in Hoshea; for he had sent messengers to King So of Egypt, and offered no tribute to the king of Assyria, as he had done year by year; therefore the king of Assyria confined him and imprisoned him. 5 Then the king of Assyria invaded all the land and came to Samaria; for three years he besieged it. 6 In the ninth year of Hoshea the king of Assyria captured Samaria; he carried the Israelites away to Assyria. He placed them in Halah [NW of Ninevah], on the Habor [Khabur River, in Turkey and Syria), the river of Gozan, and in the cities of the Medes [NW Iran].

We know that Shalmaneser was campaigning against cities near Tyre at this point, so likely he collected tribute on his way there (~725). Then Hoshea apparently (likely because the tribute was not cheap) sought support from Egypt, suggesting that Hoshea may have entered into an alliance with Tyre against Assyria, which ended in abject failure. Shalmaneser dies shortly after conquering Israel, and Israel is likely actually "carried off" by his successor, Sargon II. Assyria had deported conquered peoples since the 9th century, but Tilgath-pilester innovated the policy of two way deportations, in order to make rebellion less likely. 2 Kings 17:24 says that [Sargon] repopulated the Israel with people from Babylonian and Elam [SW Iran]. The region now became four Assyrian provinces: Dor, Megiddo, Gilead, and Samaria.

Here is a follow up question I hope might illuminate this subject... is there any serious academically known historical evidence at all, outside of religious texts, that the enslavement of Israel or anything like it actually occurred? I am finding this very difficult to google, since the results all seem to be mixed up with faith-based history.