Great question!

The information about these negotiations is quite interesting. The context is this: during the war, the Jordanian forces were perhaps the most effective. They managed to make it all the way through what we today call the West Bank, and take the Old City of Jerusalem as well as what we now call East Jerusalem generally. These were among the most important Jordanian goals, and their forces were the best-trained and best-commanded, though not the most numerous. To see the difference in success, consider how much territory Egypt managed to gain following the armistice in 1948 (Gaza, 141 sq km) and Syria (66 sq km, or almost nothing). Jordan gained territory amounting to over 5,600 sq km within the former British Mandate, a significantly more sizable gain.

The size of the territory was only one issue: the other was the way the territory was laid out. As a quick aside, when I say "border" I don't mean really a border. Many of these armistice agreements explicitly said that they did not decide borders. I'm just using it as a shorter form for armistice lines.

Unlike the Syrian border (which had difficulty in drawing an armistice for other reasons) and the Egyptian border, the Jordanian front was contentious because it ran through Jerusalem and was no straight line by any means. The result of an Israeli push to recapture Jerusalem and keep supply lines open to the fighters in the city left the West Bank looking like it had been "punctured" by a needle. There were demographic issues related to where refugees were, the length of the lines themselves and the curvatures of them, and Israeli concerns about strategic depth related to the way the West Bank leaves Israel less than 15 km across at its narrowest point. Jordan, for its part, wanted parts of the Negev for its kingdom as well, in addition to what it had gained in the West Bank.

The agreement was also pursued in greater detail, however, because of the possibility of renewed fighting. The IDF had prepared plans, if armistice agreements could not be signed, to make moves to recapture Jerusalem and widen Israel's control at narrow points. Jordan didn't want to risk that type of fighting and knew it might not turn out well, particularly since an armistice had been signed with Egypt already and the Israeli forces had proven quite capable already, and we got the result we see today. Forces withdrew to these lines with DMZs and with limits on the depths of the forces you could bring towards these lines.

But obviously, as you've noted, discussions did not end there. Negotiations occurred in spurts with Egypt, Syria, and Jordan, but none materialized into anything. With Egypt, the negotiations were not much of a negotiation at all; Egypt hinted at willingness for peace if Israel gave up the Negev, a nonstarter for them and therefore a nonstarter for Egypt. Syria had a President who offered peace for a cession of areas and half the Sea of Galilee as well as the Jordan River, but that likewise went nowhere, and he was in office for all of 4-5 months before being overthrown.

Jordan had the most serious peace talks with Israel, which is probably why your question focuses on them. Jordan feared that Israel would militarily take the West Bank if there was no peace, and as Benny Morris puts it in 1948, "it eventually did, in 1967". He figured the only surefire protection against that was a peace deal. Of course, it was Jordan's decision to join the war in 1967, but with a peace treaty he hoped that more peace could follow, and who knows, maybe that 1967 wouldn't have begun at all.

What did King Abdullah want for peace? We can't figure out too much about Jordan's intent from Jordanian records, since they're not really open to examination. We do know that Israel thought Abdullah sincerely wanted peace, and Israel said it wanted peace too, but not at any cost. It seems that Abdullah asked for Jaffa, Lydda, and Ramle, and Israel refused while demanding Latrun and parts of the territory around Jerusalem. Abdullah eventually asked instead for a 2 km wide strip of land from Hebron through to Gaza or the Mediterranean, effectively a land bridge similar to what was discussed in early negotiations between Israel and the Palestinians decades later. Israel proposed allowing access to the sea, but not with a corridor. They eventually agreed to a 100 meter wide corridor, but Abdullah couldn't get his cabinet on board at that point (if ever) with such a proposal, so the idea of peace was downgraded to nonbelligerency agreements with a term of years around February 1950. Israel and Jordan initialed one draft of one that would last 5 years, but Abdullah wasn't sure he could sell it to his people and his government, let alone fellow Arab leaders, as Morris writes in Righteous Victims, so he suspended negotiations before the deal could be finalized. Additionally, Abdullah was undercut in his negotiations because details leaked, creating widespread outcry over the possibility of peace and helping convince him that it was impossible to get it passed, as Smith writes in Palestine and the Arab-Israeli Conflict. Indeed, it seems the biggest stumbling block was not any position, but the ability of Abdullah to keep his control of the country while also getting a peace deal signed.

Abdullah, as recounted in Mordechai Gazit's "The Israel-Jordan Peace Negotiations (1949-51)" in the Journal of Contemporary History, told one US member of the UN Palestine Conciliation Commission that he was "an old man" (69 years old), one who knew his people "distrust[ed him] because of [his] peace efforts", and who felt he could "get a peace settlement if only [he] had some encouragement". That was just three weeks before his assassination. But in 1950, the Arab League had passed resolutions saying they'd expel any member who made peace with Israel. Abdullah appears to have said he was fine defying that, but only if he maintained support at home, and he didn't seem to have that. This was a reflection of popular discontent; while a monarch, Abdullah was afraid of "the street", which deposed numerous Arab rulers over the years to come. That's not to say he gave up; he continued trying to convince people in his government to support his moves for negotiations, he just repeatedly failed in his quest to do so.

Israel appears to have been aware of this dynamic, and was skeptical of the ability to seal a deal as a result; they remembered that Jordan joined the war in 1948, and Foreign Minister Sharett wrote that "[Abdullah] is not master of the situation either in relation to the British or his own government." Thus during negotiations, Abdullah did not have his council of ministers with him; he was negotiating in the hopes that a deal could be sold after the fact, rather than before. Abdullah said he would get a new council of ministers if he had to, to try and reassure the Israelis, but when the Israeli proposals that Abdullah was lukewarm to were shown to his Prime Minister, Abdullah's Prime Minister resigned, leading to Abdullah trying to find a new one and a new council of ministers. By March of 1950 Abdullah's power was thus greatly weakened because he couldn't find a Prime Minister who agreed with his policies re: peace with Israel, and following Parliament elections that resulted in resounding wins for Palestinians opposed to negotiation with Israel, any room to negotiate evaporated. Abdullah was forced to return to the person who had resigned over negotiations. At that point, negotiations died in full, as Abdullah was forced to suspend them.

Israel figured such an agreement would be pointless anyways around the same time, figuring it was Egypt who was the biggest adversary and therefore should be the biggest focus of peace. By mid-1951, Abdullah was dead at any rate, assassinated (by a Palestinian presumably opposed to peace, and that was the message Hussein took in later years when considering negotiations with Israel), and with him any view that peace with Jordan would be possible during that period.