This usage of いる is unrelated to its usual functions as a grammar element.
〜ている: progressive marker & habitual marker
食事を食べている
"I am eating my meal"
"I eat meals"
〜ていない: negative progressive marker & negative habitual marker
食事を食べていない
"I am not eating my meal"
"I do not eat meals"
〜なくている: normal V1て+V2 compounding
食事を食べなくている
"I do not eat my meal and I exist"
No one would ever say this.
〜ないでいる: adverbial form & means marker
食事を食べないでいる
"I am here (/I exist) without eating my meal." (adverbial)
"I am here (/I exist) by not eating my meal." (means)
In the case of your sentence, context suggests that it is compounding, not a means interpretation; another way to write this form is できずにいる.