Best to wait a complete season before doing any permanent planting as the ground elder is persistent stuff and may well put in an appearance next year if you missed any bits of root.

You can grass shady areas if you choose the shade tolerant variety of grass seed which is now available, as long as the shade is not very dense. A photo of the site would be helpful if you can add one to your question.

pachyphragma or pachysandra are both good though I prefer the latter as it is evergreen and a very good /tough ground cover plant that does well in deep shade.
then there are a range of ferns that will do the job for you as well as hostas. Telima, heucherella and tiraella cope in moderate shade as will many of the heucheras.

I had a similar problem and I double dug and sieved the soil before putting it back. It took me ages but the following year I only had 3 showings of plants. so it was worth the effort [border is 30mx 2.5m]

Thank you. I will photograph and post tomorrow, but comments so far are really helpful.