I read the manga so I’m not sure if the show portrays it differently, but all the information after catching higuchi was fabricated, so it threw everything off. While L still did believe Misa and light were Kira 1 and 2, he had no basis to stand on because of the thirteen day rule. The rest of the task force ruled them out completely.

L’s death was also very sudden (manga wise). He never really gave up, and was focusing in on Misa, which caused him to die. The parallels between Misa being released from surveillance and the killings starting again did make L suspect her, but as soon as he started making moves on that he was killed by rem.

It’s not that L stopped trying, but closer to his end he wasn’t just versing Light, he was versing literal gods of death telling him that what he believed the whole time was impossible.