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: this form does not exist**
> ｘ食事を食べなくている  

**〜ないでいる: normal V1て+V2 compounding & means marker**
> 食事を食べないでいる  
> "I am here (/I exist) without eating my meal."  
> "I am here (/I exist) by not eating my meal."

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.