I would focus on two important aspects the kingdoms share a lot of their culture and that they are medieval kingdoms so lets assume they are feudal states.
This means she does not have to conquer and occupy the kingdoms (with associated loss of life and resources) but only make them her vassals. To force a king to bend a knee or if he is unwilling to replace him with willing relative or ambitious lord. In turn this means they apart of the king and his immediate family nobody have much incentive to die for their king or to organize resistance after he fell. And if the stake is vassalization then even the king and his family does not have much to loose besides pride.
The shared history and culture means the people of the kingdoms will see each other as people so it is conceivable to have customs of war that limit loos of life and resources. Basically expectation that clear looser yields, that victor takes prisoners and treats them well, that peasants are not slaughtered and fields not despoiled. This could means that if she manages to beat another king quickly she can take over most his resources without depleting hers too much.
The fact that role of a king is inherited and for life means that is conceivable that at least one other king is babe in diapers and other is old and infirm. 412
The kid would have a regent with more or less skill and will to fight but the old one could refuse to give up the power without being able to do anything with that.
Next - the same reasons that prevent kingdoms to unit against P would also make it harder for the other kings to band against her.
Next as it was already said, marriage game and dynastic politics could play important role. If she can marry other king without loosing control then she could in one sweep double her power, and have a figurehead that may be more acceptable as suzerain for other kings. If she can not or do not want to marry that way, but already have children then she can merry them away converting potential enemy to staunch ally - one looking forward to inherit everything she conquer.
Finally there would be snowball effect, when she take over one kingdom she already have more power than any other alone. When she vasalize second one she is on equal level with alliance of all remaining. With four kingdoms under her control she can take on both of them at the same time and have good chance of winning. With last one remaining I would expect the king to yield and swear fealty when asked nicely.
As it was said the biggest threat is that somebody call for P before she can unify the kingdoms, but if there is a big cultural difference then the one that do may be labelled as traitor by everyone, and immediate outside intervention may mean that everybody falls in line behind strongest local player - her.
Summing up she should start with getting some allies diplomatically (or at least keep them neutral for a while), hit targets of opportunity if they show up, deal with the stronger contender first - assuming the weaker will yield if she have clear advantage, and try to get to 50%+ before her opponents organize themselves. 412