Hi. When you put drainage in you have to have somewhere to drain it to. You don't say how big the lawn is. The problem could be soil compaction or high water table (low garden levels compared to surroundings). Since you are already prepared to damage the lawn, to put drainage in, have you thought of a soak away (a deep hole filled with rubble). I have seen this work for small areas.

A soak away may not work, though, if the water table is high. If there is a layer of clay, with faster draining soil underneath, piercing that layer every 2 meters with a soil auger and filling the holes with sand may be enough. If a hole dug there quickly fills with water--without the help of rain--it may be a high water table, and making an ornamental pond may be the only practical answer. If it is simply a problem with tight clay soil, aeration, an application of lime or gypsum (depending on the pH) and regular applications of compost, will eventually deal with the problem.