I don't think we need to assume L'Rell's grip of the Empire's throat was any less or more solid than that of previous or later leaders. Bosses come and go; they don't remain in power due to anything as fleeting as campaign promises ("I will not blow up this planet if you obey me"), but through the classics: finding powerful cohorts who agree to let the current clown reign so that they can have all the niceties of being on the side of the boss without having to be the boss.
L'Rell would have plenty of cohorts standing by, as her House is one of the Great ones, supposedly with creepy tentacles in the groins of all the others. And there'd be plenty of loose muscle around at the time, too, as Kol's alliance-happy allies would now be free to ally with somebody else for their alliance needs.
Why should anybody feel the need to depose L'Rell at once? Klingons love having leaders. But having weak leaders is better than having strong ones: if you depose a weak leader, you make little impression, but if you let a strong one stand, he may eventually come to do some actual leading - and Klingons don't really love being led, by their leaders or by anybody else.
Leadership is just a great excuse for doing things - hence, T'Kumva's initial success. Everybody already had their fun with the UFP war, and (as established in the strategy meeting) nobody actually cared about winning. L'Rell is an excellent excuse for having peace and showing off your loot. When the War Tooth begins to itch again, then it may be time to change leaders. But before that, somebody will have to do something about the volatile basements of Qo'noS. Why not let the current leader handle the chore?
Timo Saloniemi