The rules very from state to state. But here in Texas, if you withhold part of the rent without getting written consent from the landlord, it can be considered not paying rent. Which could lead to eviction. THis is why many landlords will not take partial payment for rent.
As far as the part, if the dishwasher is under a warranty, your landlord may be telling you the truth. When my warrantied dishwasher broke, I was told the part was back ordered for six months. I could buy it other places, but it was very expensive.
As annoying as it is...and I KNOW how annoying it is. We waited about a year to get a replacement dishwasher. I do catering and cake decorating. Hand washing is NO fun! As annoying as it is, a dishwasher is not something a landlord is mandated to fix. A dishwasher is considered a matter of convenience.
I would contact the manager again and ask if the part could be ordered from one of the sites you have found. I think you are right about not wanting to be aggressive. I am much more willing to work with tenants in our rental property that want to work with me than those who go on attack mode. Ask her if you can order it and withhold the amount for the part from next month's rent. If she agrees, you are covered. I would be very careful about taking the money out without permission. If she chose to pursue it, you could be in legal trouble. Hopefully she will agree. GL!