I went to this website to get a definition of irony, and will focus on this one:
2: a) the use of words to express something other than and especially the opposite of the literal meaning
When I think of someone who is successful, my initial thoughts are a well-paying job, and a life of relative ease and comfort. Ditch digging is hardly the first thing that comes to mind.
In that sense, I think there could be a trace of irony in the bolded sentence. However, I wouldn't characterize it as dramatic irony; I'd regard it as sarcasm. By dramatic irony, I'm referring to this definition of irony:
Incongruity between what might be expected and what actually occurs
There isn't enough context in the scene to call that situation ironic. Sarcasm, however, refers to the use of words to mean something opposite of their usual definition, normally in a caustic tone:
harsh or bitter derision or irony.
a sharply ironical taunt; sneering or cutting remark.
In short, sarcasm is a form of irony, and the thought reads as sarcastic to me. So, if I read the paragraph by itself, I might perhaps detect a smidgeon irony there – though it's hardly the best example of an ironic statement.
With additional information – namely, that we are reading the thoughts of a depressed and suicidal woman – what little irony may have been read into the passage seems to evaporate. In that context, I see despair, not irony.
Incidentally, it's not always easy to judge if there's irony in a passage. In fact, you can even submit a passage at a website, and let people vote on whether or not they find that particular example ironic. My guess is that most irony connoisseurs would cast votes on the NOT IRONIC side of the spectrum.
It's also worth noting that the Is It Ironic? website doesn't even use two buttons (IRONIC vs NOT IRONIC). Instead, there is a 1 through 10 rating system, which includes a section labeled "Tricky - Not Sure." Irony isn't always so easy to judge and detect. Maybe that's ironic, in a way?