This is a good question, but I want to start by clearing up a couple of things. Firstly, some definitions: 'Canaan' is usually used to refer to the area from the Eastern Mediterranean coast to the hill country that's farther inland, and often comes with an implied sense of 'not Israel and Judah'. I'll use it here to mean everything in between Egypt and the Hittites.^1 The only significant collection of contemporary texts we have for this area are from the city state of Ugarit (now Ras Shamra, in the north-west of modern-day Syria). There are some inscriptions from the various Phoenician cities and a couple from the kingdom of Damascus, but evidence is very, very scarce.

Secondly, the conquest of Canaan as described in the Bible has practically no evidence to support it and is probably a complete invention.^2 Major problems with the biblical account were first raised when the archaeological record from a large number of sites around the area didn't seem to add up to what we had been expecting to find: contemporary destruction layers in the Late Bronze Age (around 1100 BCE or so, when the story is set), and an abrupt change in artefact styles (pottery, architecture, jewellery) and cultural practice (burial style, traces of rituals, diet). None of these were there, and in fact what we did find was continuity between Late Bronze Age and Early Iron Age. In other words, the native population probably wasn't replaced with two million or so Israelites coming from Egypt.

Thirdly, Jericho is actually one of our most important case studies when it comes to understanding the history of Israel in Canaan. The really big excavation of the city (focusing on Neolithic and Early Bronze Age Jericho, but including a broad survey of the entire chronology of the site) conducted by Kathleen Kenyon in the 1950s found some pretty remarkable things.^3 To summarise, she discovered that the city had had significant fortifications in the Middle Bronze Age (roughly up to 1550 BCE), but that these had been destroyed completely during that time. Of course, this is much too early for anything remotely related to a political entity 'Israel'^4 to be relevant; even the most conservative scholars date the reign of David^5 to the late-11th or early-10th century BCE. Jericho was only a minor settlement during the Late Bronze Age and Early Iron Age, and was even mostly abandoned for several centuries during this period. It wasn't until the 9th century BCE that Jericho was properly rebuilt.

Now, to actually get to your question: who ruled Jericho before it was destroyed? Whoever it was, they didn't really leave us much to work with. According Kenyon's interpretation of the evidence, Jericho became a city of some importance - and perhaps with some degree of independence - like many other major Middle Bronze Age sites (including Ugarit) in the first half of the second millennium BCE. Their cemetery was extensive, and the elaborate nature of the burials there suggests that the rulers of the city were relatively well-off. We know from Egyptian scarabs found at the site that they traded with the Egyptians. Kenyon didn't find much in terms of religious worship, but geographically speaking it is likely that they shared the general Canaanite pantheon with the other cities in the region (Ba'al-Hadad, El, Anath, and Astarte being their most important deities).

I am not aware of the city being mentioned in any extra-biblical texts (neither from Egypt nor from Mesopotamia), but I am happy to be corrected if anyone else has an example.

I hope this helps! Please let me know if I can expand on or clarify anything!

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^1 Scholars are usually pretty inaccurate with the term, so depending on preference some will include or exclude the Phoenician city states; some even only use 'Canaan' to refer exclusively to the area that would become the territory of the kingdoms of Judah and Israel.

^2 see Davies, In Search of 'Ancient Israel'; Finkelstein and Silberman, The Bible Unearthed; also Dever, What did the biblical authors know and when did they know it? (a little outdated now, but still a good, neutral survey).

^3 See Kenyon, Excavations at Jericho, several volumes spanning the periods 1954-1958 and 1960-1983.

^4 The Merneptah Stele, dating to the 13th century BCE, refers to something resembling Israel, but it should be seen as a name for the area rather than a political entity.

^5 David's historicity is a can of worms I really don't want to open here, but even assuming he existed and ruled over a united kingdom he's a good four centuries too late.