unless you solve the drainage then these unwanted plants will only re-establish themselves. The feed and weed range with moss killer will get rid of the moss and may get rid of the reeds. You may be able to use a daisy grubber tool to get the reeds out unless it is too big an area.

Plus, a weedkiller to treat lawns probably won't touch the rushes and reedy growth. In fact, it may promote it.

May sound pessimistic but it would probably be unlikely you could establish a good reed free lawn under those conditions. Have you tried very regular mowing on a fairly long cut? (a short cut encourages the moss and discourages anything that wants to grow tall)
It sounds ideal for a good bog garden though.