Why didn't the experimentation for the 13-day rule push through? In the episode where L dies, he mentions wanting to test the 13-day rule. If the 13-day rule is disproven, Light and Misa become suspects again. L dies and then apparently everyone seems to have forgotten about the test. What happened?

I seem to recall most of the task force disagreed with L and then when he died, the majority opinion prevailed.
For them to test the 13-day rule the Death Note has to be used. the Task Force's disagreeing with L was over the fact they would have to sacrifice someone for an experiment.

The Japanese Police were already a bit iffy of L since the Lind L. Tailor incident where L had a condemned criminal sacrificed just to test how Kira kills and where they may be, and the Police and the Task Force have to keep the moral high ground and not go around killing people for tests/experiments, especially with the Death Note.

Also remember that the Task Force didn't learn about the Death Note or the fake rules until quite some time after Light's and Misa's confinement where "Kira" was a member of the Yotsuba Group, all the while Light was handcuffed to L. to the Task Force this was more than enough to clear Light's and Misa's names without stooping down to Kira's Level, L on the other hand can see himself on the same level as Light as per what he said in Death Note Relight 2: L's Successors


  There are...many types of monsters in this world: Monsters who will not show themselves and who cause trouble; monsters who abduct children; monsters who devour dreams; monsters who suck blood, and... monsters who always tell lies. Lying monsters are a real nuisance. They are much more cunning than other monsters. They pose as humans even though they have no understanding of the human heart. They eat even though they've never experienced hunger. They study even though they have no interest in academics. They seek friendship even though they do not know how to love. If I were to encounter such a monster, I would likely be eaten by it. Because in truth, I am that monster.


Source: L - Quotes (9th point)

take notice of the lines i emphasized. it forebodes his death but also describes the kind of person Light/Kira is, and that L himself is just like that so L would have no problem "testing" the Death Note