Good Afternoon,
Lily of the Valley is not easy to eradicate.
Here are some options. You mention other plants. If your other plants are close to the lily of the valley, you will have to be especially careful if using an herbicide. 
While this plant may look like its harmless, it is fatal when ingested. Some also asked if it is dangerous even to touch. Luckily, you won’t die just by merely touching it, but it is not safe either as this increases the chance of exposure.
If you want to kill this plant for good, then you need to dig up its rhizomes from the soil. Any tiniest portion of a rhizome is enough for the plant to regenerate.

1. Wear a pair of gardening gloves before you start the process. As mentioned, all parts of the plant are poisonous, and you want to avoid its toxins to stick to your hand and accidentally consume it later.
2. Dig up the section where the lily of the valley grows using a spade. Make sure you get all the roots and avoid leaving even the slightest rhizomes in the soil. The roots of the lily of the valley usually spread horizontally, so expect to extend sideways when digging up.
3. Get your rake and proceed to raking the area to spot loose or broken rhizomes if there are any.
4. Cover the whole area with old newspaper and cardboards in about 3 to 5 layers and secure it with heavy items on top. This covering will prevent the loose rhizomes from regenerating by cutting its source of air, which cools the soil down. In the long run, the temperature underneath the layers will increase and eventually kill the roots of the plant.
   
5. Wait for approximately 6 months before you cultivate the area again and prepare it for the next plants you want to grow. The waiting period is crucial to ensure that all the rhizomes of the lily of the valley are killed and destroyed. It is also best to do it in the dry months when there is lesser chance of rain.
6. The newspaper, cardboards, and mulch you use will also break down and serve as added nutrients to the soil.

Sadly, the best way to kill lily of the valley in a short period of time is by using a non-selected herbicide, specifically the Round Up herbicide. This kills everything it encounters, which includes all the rhizomes of the lily of the valley.

Follow steps 1 to 3 from the earlier method and spray the recommended amount of herbicide on the lily of the valley plants only. Wait for a few hours and re-apply again. After which, you need to wait for 3 weeks and rake the dead plants in the area.

Since you basically kill everything in the soil bed, it is necessary to re-condition the soil again before you proceed to plant new seedlings.

Right after clearing your garden from lily of the valley, it is not advisable to plant edible plants right after. You may, however, plant shrubs and trees. Be sure to read and follow the directions with great care.

I hope this helps.  Thanks for using our service.