Some lawn mowers such as the Craftsman 37705 have a washout feature. It is essentially a garden hose fitting on top of the circular blade cover. This allows you to pump water into the blade assembly while running the engine. That action is designed to flush dirt and grass debris right out.
After sharpening both sides of the blade place it on the kits cone-shaped balancer. If one side dips below the other continue sharpening it until the blade rests level. Return the sharpened blades to the mower and reattach. If you have a torque wrench its a good idea to use it here. Often you need to tighten the blade mounting bolt/bolts to a specific tightness level. Typically measured in pound-feet your instruction manual should list these specs. For example my personal machine the Honda HRR216VKA between 36 to 43 lb-ft of torque is required.
There is something Zen about spending an hour or two each week pushing the mower around the yard on a sunny day. The drone of the motor drowns out your thoughts as you work on a task with a clearly defined objective: Making your lawn look its best. Of course the deciding factor between an almost meditative experience and one filled with frustration is often the quality of the lawn mower you have.