It was correct not to perform service on the nail - if the nail bed and/or matrix is exposed, performing service on such nail may cause even greater problems.
You should advise your client to wear gloves while doing dishes, cleaning, etc. - to protect this nail unit in particular, as well as, all the rest from exposure to chemicals and dirt. If the client is experiencing pain, swelling, redness and other symptoms associated with inflamation within this nail unit she should go and see doctor.
What you and your client should expect to happen with nail unit? Most likley nail plate will completely seperate and come off, however, new nail plate should be formed by nail matrix in mean time, so within 8 months her nail should be as new. Yet, the hole in nail plate worries me - as previously I mentioned it might be sign of permanent nail matrix damage, which means, that nail matrix in this area won't be forming new nail plate (you can google more about nail matrix damage for some scientific papers to understand better). In such case (permanent damage) there really ain't anything that can be done - maybe (depending on damage) it could be sculpted by gel to visually make it look good/better.
Why this happened? Well, if client is hard on her hands/nails and her nail matrix is genetically located near proximal nail fold that may have made it more prone to damage, which you couldn't know... so even slight manipulation to this area may have huge impact. In future, for this client, I suggest you use orangewood stick instead metal cuticle pusher and just scrape off cuticle of nail plate, but do not push it back.
Don't blame your self - if what you said is true (you were gentle & client is harsh on her nails), yet I understand it is hard. Appologise to client, maybe offer service once for free. Explain, that this is not typical to happen and this is your first experience with something like this ever.
Sorry for typos - english is not my first language and I am in great hurry. Hopefully you understood everything I wrote.