You have a formidable task on your hands and unfortunately there's no easy way out, short of moving. The only ways to get rid of wild onions/garlic is to either dig them up, or use herbicide. However, timing, technique and diligence is essential for success. Here is a website which goes into more detail than I can do here.

http://freeplants.com/get-rid-of-wild-garlic.html

I use a combination of techniques. digging as and when I can, dead heading the flowers and spraying with roundup at the time of flowering so the weedkiller can be taken down to the bulb and kill it. it isn't a quick job but it can be done.

welcome to GoY too :o)

Before you put any weedkiller on just try putting a leaf in a ham sandwich - yum!