What happens when the party your suing defaults?

Details of case:
Knee surgery twice and spinal fusion surgery which resulted from falling down steps at a workplace here in nyc which now because it’s been almost 2 years they let you go(Fire you) due to missed work. Worked for the city of NY.

Reasons I am asking is because usually I see lawyers go after the parties like owners and find out who their insurance companies are and contact the insurance if they cannot get in contact with the owner of the property but the lawyer my father has I don’t see doing that. which is weird and we maybe thinking about switching depending on how you respond. Or maybe we are being hasty. I don’t know, please advise me.

Also, as of right now workers comp are paying for everything in terms of his surgeries and pay every 2 weeks. Fortunately NYC didn’t own the property and it was being leased to the city through a corporation. Their steps before they fixed it after the incident was really really in bad condition.

And finally, what do you think a case like this is worth?

Awaiting your response, thanks!