If lev melech beyad Hashem, then do we not show appreciation for good things they do and blame them for the bad things they do? If that’s the case then how can we blame even Hitler ym”sh? What about free will?
The verse “Lev melachim byad H-shem” (Mishei 21-1), does not mean that a king does not have bechira at all, because as you mention, no king would be accountable for his actions. However we do see that every Jewish king, Saul with Amalek, Dovid with counting the Jews, Shlomo with marrying the daughter of Pharaoh, and all the other Jewish kings, were all held accountable for their actions. Even gentile kings were held accountable for their actions, such a Nebuchadnezzar, Koresh, Antiochus, etc. Even regarding Pharaoh in Egypt, the Midrash says that he had bechira for the first five makos, and only afterwards was his bechira removed. We see from this that even kings do have a level of bechira, and therefore they are accountable for tier actions.
The meforshim explain, (see Chasam Sofer Parshas Vayigash who brings this idea in a different context), that this verse doesn’t mean that they don’t have any bechira at all, rather regarding davening to H-shem to take away a decree or plan of a melech, that it is easier to remove the plan of a king then the plan of a regular person. This is because the heart of a king is in the hands of H-shem, therefore by davening to H-shem it is easier to get the king to have a change of heart than a regular person, since the regular person’s heart is subjected to his own bechira. This idea is also brought in Yaaros Devash (at the end of Luchos Haedus) regarding the gemora Megillah 11a that the decree that the decree in the hands of Haman who wasn’t a king was more difficult to abolish because he wasn’t a king.