As Caspar has alluded to in a comment, a child theme descends from a parent theme – you wouldn’t add a pre-existing child theme on to a parent theme the same way a parent human is needed to make a child human 😉
The theme you are working on now is a parent theme, whether it came from another theme base originally or not.
You have two ways forward:
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You can continue to treat this theme as a standalone parent theme. This is ok, because it’s not going to be updated from the WordPress theme repository or edited by anyone else at this stage. Or:
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You can create a new child theme and use this theme you have inherited as the parent theme. This is not a bad option if you want to retain the existing theme and avoid overwriting any of it with your changes.
I would recommend option one in your circumstance. Given the theme is broken already, you’re unlikely to want to remove many of your changes (but, always wise to keep a backup of the existing theme anyway!). But secondly and probably more pertinent to your situation, if the theme is built badly in the first place it’s highly unlikely it would have been built with with child theme extension in mind. While you can probably override most of the CSS without an issue, you may come across functions that have been defined by the theme that you want to change, and just can’t.
In summary, there is no rule that you must always develop on child themes. It’s only a good practice when you’re building off a well-established, frequently updated public theme. Which isn’t the case with what you’ve inherited!