Under normal conditions, the investigation of any incident which involves a Senate member - including assassination attempts - is the sole jurisdiction of the Senate Guard. The Senate Guard is also responsible for protection & ceremonial details, and can often be seen standing at attention in their vibrant blue armor. The guard answers directly to the Supreme Chancellor of the Republic.
Under special circumstances, the Chancellor can ask the Jedi to handle an investigation instead of the Senate Guard. This is usually done in high-profile corruption cases, and it should be noted that (internally, at least) many members of the Senate Guard see it as a slight insult. This would probably indicate that it's not done often.
In the case of the assassination attempt on Senator Amidala seen in Episode II, Palpatine directly involved the Jedi - thus replacing her normal Senate Guard detachment with Obi-Wan Kenobi & Anakin Skywalker. By extension, they also became involved with the investigation. As the audience, we know that this was done to further his own plans. Without that knowledge, it would seem that he was doing it because of his friendship/concern for Amidala - the senator and former Queen of his own homeworld.
In the case of the Jedi "assault" on Palpatine, any investigation would be handled by the Senate Guard, under the direct supervision of Palpatine himself. Obviously, given such a situation, Palpatine would be free to influence the investigation however he wanted. Therefore, it would be easy to blame the burns on something else, such as calling them blaster burns from the Senate Guardsmen who came to his rescue. For that matter, it would be easy to falsify the autopsy results altogether, making the "official" cause of death whatever was needed.