This usage of いる is unrelated to its usual function as a grammar element.
"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.
"I am not eating my meal" (progressive)
"I do not eat meals" (habitual)
?? "I eat my meal and I am not here (/I don't exist)" (conjunction)
Again, reading #3 is a terrible way to read this sentence and is essentially wrong.
?? "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.
"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.