That would only work if they can find the hole. According to the article, it often doesn't show up on imaging, and even if they know roughly where it is (as in the case of a lumbar puncture gone wrong), it's still incredibly difficult to pinpoint, and patching it usually involves trial and error.
It wasn't completely clear from the article, but it sounds like often they don't ever really pinpoint the hole, even when they are able to successfully patch it. The patching process seems more like "spray and pray": do some patching (she had 22 patches done, I think I read), and once it seems like the leaking has stopped, they assume they got it. Not sure if this is true, but that's the impression I got.