No no, DO NOT put the vase in the dishwasher!!!! "Cycled" means letting it sit for some weeks/months to allow nitrogen-cycling bacteria to colonize and proliferate, so that there gets to be a big enough population of nitrifying bacteria in the tank to convert the fish's ammonia output into nitrite and then into nitrate, the latter being far less toxic to fish than ammonia or nitrite.
However this is not really an issue in a container where the water is changed frequently, as it sounds like you are probably doing with this vase setup (?). If you are *not* changing the water at least weekly, preferably a couple times a week, then it could in principle be part of the problem, although by now the vase *would* be cycled-in reasonably well so it would not be your *current* problem.
Really though, get a tank, you can still grow bamboo stickin' out the top if you want, and the betta will be much happier and healthier.
Good luck, have fun,
Pat