Use a garden fork to spike the lawn to a depth of 4-6 inches at 6 inch intervals to improve the drainage. ( It is very hard work so do a bit at a time)

You can also buy a tool which has hollow tubess which leaves hole in the turf wide enough to fill with horticultural sand which should create more permanent drainage

Susan, you have my sympathies, I have the same problem in areas of my lawn. In my case there is soil compaction under the topsoil and as I have clay soil, it is quite difficult to sort out. Anchorman has given you the answer really, it's hard work, but it is worth it. I would definitely go for the horticultural sand if you can get the tool...there are also companies that specialise in doing this work if you can't do it yourself.