Answer: I am so sorry you are having this problem, I know this plant can be so invasive especially in moist rich soil. Digging it out thoroughly and/or using a herbicide containing glyphosate are the ways to try to get rid of it; stop fertilizing and stop watering to slow it down as well.
The herbicide needs to contact ample foliage so it can be translocated to the roots, so you have to let the loosestrife grow a bit before spraying it. As you have noticed, the roots are aggressive and deep, so sometimes an herbicide is more efficient but it too may take several applications. Be sure to read and carefully follow all of the label directions.
If this were my garden, I would either dig or give up on the perennials in the worst affected area. You would dig up the perennials you want to keep and wash the roots clean -- in other words bare root them -- then triple check that they were free of the loosestrife roots before replanting anywhere in the garden. In the meantime I would probably put them in containers temporarily to make sure not to accidently introduce the loosestrife elsewhere.
Or, you might consider sacrificing the more easily replaced perennials and do an overall treatment of the area. Then dig a perimeter trench and keep a very careful eye on the surrounding garden to make sure you didn't miss any. It is much better if you can get after it as soon as possible rather than delay and allow it to continue to get a hold. If you do this, you could probably root some tip cuttings from the perennials you wanted to keep so it isn't a total loss.
I wish I could be more encouraging. Good luck!
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