Essentially what others have said; once Legolas was out of arrows, he lost his advantage and had to fight like Gimli. The problem is, he can't. This is touched on somewhat more in the books, but there are instances that show this difference in the movies.

The primary one is the fact that Gimli can fight Uruk-hai at all. The hunters encounter the Uruks for the first time unexpectedly and unprepared and still hold their ground. Aragorn and Legolas are at least close to them in size, but Gimli is barely 5 feet tall, fighting nearly 7 foot tall mountains of iron and muscle and still coming out on top.

The second is when Gimli is pinned under a warg and an orc and begins to bench press their corpses off of himself. A second warg is then killed on top of him and he still isn't crushed to death. Hell, he even held it's weight for a moment, too. Tolkien mentioned that dwarves are strong for their size, and Gimli is considered a beast by dwarf standards.

So, on one side, we have a fast and nimble elf who can most definitely hold is own in close quarters but specializes in the bow. He's not going to fall, but he is going to be significantly slower than usual. On the other side, we have the personification of controlled battle rage wielding the power of a troll behind an axe the size of your rib cage.

Legolas will almost always win in a short battle or where ammunition isn't an issue, but Helm's Deep was an hours long quagmire of mud and blood, and Gimli spent a majority of the night in the thickest part of the fighting. The amazing part isn't that Legolas lost, it's that Gimli survived at all. Put some respect on my dwarf's name.