As it's explained later on, another nation's army would result in uniting the Houses to fight the invaders. It's in the bones of Andor. Morgase just could not seek the help of another kingdom (or foreign House, not that the Damodreds seem to be in any shape to provide her an army - their enemies would have taken advantage of this in the civil war). None of them would have provided her an army without important gains in trade or territory, and none of them would have remained around if it meant starting a war in Andor against allied Houses. If Morgase showed up with Cairhienin, or Murandians some of the Houses would have fought her before turning to fighting Gaebril, and it's not only Morgase who would have been finished, but her House too (the civil war would have happened sooner than later, once it became official Morgase had vanished. With her escape, he could no longer pretend she ruled and he could pick no other puppet, which was her little victory over him.).

Morgase needed a big army at her back for the civil war certain to come (it was a matter of months, the time for the Houses to negotiate and mobilize). She had to make a decisive move to retake the capital, and hope that when they saw her fight Gaebril, some of the Houses she had alienated would rally to her. She must have known her rule was in jeopardy and she might have no choice but to abdicate in the end - it's mostly Elayne's right to rule she intended to preserve, I think.

Her best option was to mobilize an Andoran army, but she needed a General. This option was removed from her when she couldn't find Bryne. Her compulsion prevented her from seeking the second best (of a series of all bad options) option, which would have been to seek the Amyrlin's help. Sisters would have been sent to negotiate with the Houses Morgase had alienated to make them help Morgase to oust Gaebril, and negotiate at which price. Morgase didn't know, but Elaida even had additional personal reasons to throw the weight of TV on her side (her foretelling).

Given that she could hardly achieve this with mercenaries (only a fool would attempt this), that left Morgase only one choice: the Children of the Light. She was well aware Niall was a rational man and fine political animal. She knew he had long wanted to increase his influence in Andor and undermine that of Tar Valon - and the deal she wanted to offer him did just that. He would not refuse her, it was only a matter of what price he would ask. If she played her game well, she might have Galad at her side, and as many Andoran Children as possible, when she returned to Andor. For Niall, Morgase might even set a positive precedent in calling to the Children to the help of a rightful ruler, the way a ruler might have sought AS help. An army of Children would not please the Houses, but not the same way a foreign national army would. They would be seen as mercenaries, not invaders, though they would worry a lot about the price Morgase paid for them. If they didn't force her to abdicate once Gaebril was gone, she looked for decades of work to undo the damages, but she knew all this.

In her position of weakness, the last thing Morgase could do was to go to Pedron Niall and beg. She had to make no offer, she had to let him know what she wished by askign someone else, and let him come to her and make her his offer. It was the only way she could regain a little of a bargaining position. That's why she went not to Niall but to his puppet Ailron. Niall saw right through her game, and let her stew. He demolished her bargaining position, and when she had none left, he did give her what she wanted.

What Morgase didn't foresee, and she can be excused because not even Niall and Balwer saw this coming (everybody thought Niall had those elements under control), is that her presence and the imminence of a deal with her would spark the rabid fanatics to move against Niall.