It's a little more complex than one owner of the domain/hosting account. The domain dooranddock.com is registered to (and the admin contact is also to) Passport Door Systems, Inc. That's who it sounds like you are working for. And that's good for this situation, as it shows at least one person on the whois record is the related to the domain name itself and the owner of the actual website and business.
But the technical contact is at digital-info.com ; is that the old designer? They must have the login/password to the NetSol account? That account controls the domain registration and hosting. And that's what you want to do: access the NetSol account.
Once you can get into NetSol, you can move the site to your own host and remove others from the domain registration. Or, digital-info.com can turn over the logins and let you take control of the NetSol account and billing.
It's your choice, but a simple FYI letter or email from an attorney to the old designer - and/or to digital-info.com - can work wonders resolving domain ownership and access to a hosting/domain registration account. (And I'm not an attorney and this is not to be construed as legal advice, either.)
The site of the branding designer of the site - prehistoricgeek.com - is a dead domain. The designer's agency digital-info.com is live.
If the old designer is simply unavailable in any way, and digital-info.com has nothing to do with them anymore, you need to talk to NetSol and try and talk them into deciding ownership of the hosting account and the domain in your favor.
That could be a problem. But tell NetSol the site is hacked and that might help you get access; who knows?