The bad - Lily of the Valley spreads aggressively by underground rhizomes and will continue to spread without intervention.  While tough to control it can be done but only with perseverance.  Digging it out is one option that works 100% of the time, but only if all of the underground stems and rhizomes are removed.  Leave any part in the soil and it will come back.  If you want to get rid of all of it, cut it low to the ground with a string trimmer and then cover it with black plastic, cardboard, landscape fabric or newspaper weighted down and left in place for the rest of the growing season. If you wish to keep some of it, digging will be about your only option.  Lily of the valley can also be killed with glyphosate, the best known being Roundup.  But it will take more than one treatment.  Know however that glyphosate is a non specific herbicide, meaning it will kill most everything it touches, if there are other desirable plants nearby that you wish to keep.  Finally once the leaves are fully grown you can just pull them as they snap off easily.  This from personal experience.  They will all be back next season of course but won't return this season.   

The good - Flowering, fragrant Lily of the Valley can be a very nice addition to any shade garden in the right place when kept controlled.   

Good luck!