My dog was hurt by the Vet, do I have a case worthy to pursue?

To make a long story short, I took my dog to the vet for a routine follow up for an ear infection and ended up having my dog choked by the Dr. This choking led to fluid on her lungs, internal bleeding and 4 days in ICU clinging to life. Although the vet has paid the ICU bill and has offered their wellness program to my dog, they have refused FULL responsibility for the financial, emotional and mental distress they caused me and my family. For example, we informed (vet) about the loan we took out to pay the ICU bill before they finally agreed to pay it (admittance of guilt), but they continue to not make the family whole in any way. So again, once I threatened to sue them, they have only agreed to offer my dog their wellness program, but deny responsibility of the distress caused towards the family. Our proof of claim was also shared with the Vet, in an attempt to settle out of court, but to no avail.