This usage of いる is unrelated to its usual function as a grammar element.

**〜ている**
> 食事を食べている  
> "I am eating my meal" (progressive)  
> "I eat meals" (habitual)  
> ?? "I eat my meal and I am here (/I exist)" (conjunction)

Reading #3 is never used because no one would ever need to say that. I included it only to show that the て-form does normally perform a conjunction function, it's just very marginal here.

**〜ていない**
> 食事を食べていない  
> "I am not eating my meal" (progressive)  
> "I do not eat meals" (habitual)  
> ?? "I eat my meal and I am here (/I exist)" (conjunction)

Again, reading #3 is a terrible way to read this sentence and is essentially wrong.

**〜なくている**
> 食事を食べなくている  
> ?? "I do not eat my meal and I am here (/I exist)" (conjunction)

This form is never used because there's never a need to say this.

**〜ないでいる**
> 食事を食べないでいる  
> "I am here (/I exist) without eating my meal." (state adjunct)  
> "I am here (/I exist) by not eating my meal." (instrumental adjunct)

Reading #1 is `Lit. "I am here, in the state of not eating my meal."` (It's reminiscent of the stative function 〜ている often performs, but it's slightly different — here, `いる` is actually still a full-fledged verb and you can't drop the "exist" meaning.)

---

In the case of your sentence, context suggests that it's a state adjunct, not an instrumental adjunct; another way to write this form is できずにいる.