The Caliph was overwhelmed by confusion:

He was inclined to accept the proposals of Mu’ayyid Ad-Deen Al-‘Alqami, for he was not strong enough to enter a war. But his mind supported the advice of Mujaahid Ad-Deen Aybak, for there was no opportunity of peace with the Tatars throughout their history. He also knew that rights would not be granted in so much as taken (by force).

The Caliph was confused, and in the end, he resolved the matter.
He, thanks to Allaah, listened to the words of logic. He decided to strive. But he was reluctant, weak, lenient and easygoing; these were all attributes with which Jihaad would be pointless.
Jihaad is, by no means, a decision to be taken lightly. There is no Mujaahid by chance. Jihaad requires preparation, education, sacrifice, and a long journey on the way of faith. Jihaad is to rise higher until you reach the culminating peak of Islam. But, anyway, let us strive, by way of experiment. The Caliph gave permission for the use of the army, which was the first time in his life.

A feeble brigade of the Abbasid army led by Mujaahid Ad-Deen Aybak set out to face the formidable army of Hulagu. As soon as the Abbasid army set out and started to prepare themselves to meet Hulagu, Mujaahid Ad-Deen received the news about another Tatarian army coming from the North, i.e. the army of Baiju coming from Europe across the territories of Turkey and North Iraq. This army had previously passed through the Iraqi territories East of Tigris until when it came to Al-Mawsil, it crossed the Tigris to its West side and marched through the area between both rivers of Tigris and Euphrates until it became at a distance of about 50 kilometers from Baghdad.

By the time it arrived in this region in the North of Baghdad, the news had reached Mujaahid Ad-Deen Aybak.
Mujaahid Ad-Deen Aybak learnt that if this army arrived in Baghdad, it would encircle it from the North and West, and thus tighten the blockade on the Islamic capital. So, Mujaahid Ad-Deen Aybak decided to quickly change the direction of his army toward the north, in the area between the Tigris and Euphrates, in order to meet the army of Baiju. He actually met the Tatarian army at Al-Anbaar, which witnessed, six hundred years ago, a historic victory at the hands of the momentous hero Khaalid ibn Al-Waleed, May Allaah be pleased with him.

But at that time, victory, unfortunately, was not the portion of the Muslims. Baiju seemed to have been more acquainted with the region than its inhabitants. He started to give the false impression of withdrawal, and was able to gradually draw behind him the Muslim army until it was brought to a region of swamps near the Euphrates. Then, he sent the Tatarian engineers to break the dams held on the Euphrates in this region, in order to cut off the line of flight for the Abbasid army. Then, Baiju besieged the Iraqi army and started to execute massive genocide against the Muslim army; and Mujaahid Ad-Deen, along with a very small brigade, was able to withdraw southward, alongside the river and return to Baghdad, after the greater part of the Abbasid army had been ruined in the region of Al-Anbaar.

This painful unequalled battle took place on Muharram 19th, seven days after the appearance of Hulagu in front of the Eastern walls of Baghdad. Wasting no time, Baiju moved forward and reached Baghdad from the North just on the coming day (of the battle), and turned round Baghdad to besiege it from the West. Thus, Baghdad was caught between both jaws of pincers: Hulagu from the East and Baiju from the West. The situation became more difficult, and the blockade was tightened upon the capital of the Islamic caliphate.

The Caliph, son of caliphs and sultans, never imagined that he would be put under such a blockade, to the extent that his mind stopped thinking.
Mu’ayyid Ad-Deen Al-‘Alqami came to avail himself of this opportunity (saying): "O caliph! There is no way but to sit with the Tatars at the negotiation table."
But the Caliph knew well that if a very strong party sat to negotiate with a very weak party, there would be no negotiations between them, but the result would be surrender (on the part of the weak), according to which the defeated should accept all the conditions imposed by the victor without the least modification or objection.

Lowering his head, the poor Caliph accepted surrender, I mean negotiations.

https://en.islamstory.com/index.php/en/artical/20765/confused-caliph