Like chica I have masses of ivy in my garden and have had no problem growing spring and summer bulbs and perrebials alongside/underneath it. Ivy is a great source of food for insects and specially some moths.
Some of the stone dykes around my garden are completely covered in it and is popular with wrens and other small birds so I leave it largely to itself except for cutting back the creeping stems about this time of year whilst I can still reach them and then leave it to develop berries for the winter.
If it is possible I'd keep the ivy - I've never heard about it having poisonous fluid in it's roots.
I don't know what the mile a minute plant is but years ago I had a Russian vine that was called the same thing (otherwise known as triffid!) - it grew like crazy and covered a home made pergola in one season and was a mass of white flowers.