...you realize how fragile it is that with a blink of an eye of a
broken toilet, a sick child, a sick parent, that that balance is
'a blink of an eye' free dictionary
blink of an eye - a very short time
In this excerpt 'a blink of an eye' means a very short time. The narrator has realize[d] how fragile the work-family balance is, because she has a 'broken toiliet, a sick child, [and] a sick parent'. These are all events that happened quickly (emphasised by the way they are rattled off, alongside commas from a textual point of view) and the fact of course that we have the use of the phrase 'blink of an eye' which is forming part of the description.
I think the phrase is a particularly good choice here as the metaphor works very well, you can imagine her blinking her eyes around her as all of these random events take place, breaking her fragile 'work-life balance'.
How fragile, simply means it is not very strong, in this case her 'work-family balance' can be toppled very easily by these random events.