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I think that's the cleanest way. Continuous gutter, metal. Do they make a plug patch I could seal up? Stopping by the store again on the way home and will look.
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You can just rivet in a piece with some sealant. But it'll look crappy
Honestly I'd just replace that section of gutter with a new piece and put the outlet in the right place to hit the other side of the house
Eta: I'm not sure what the initial problem is for sure. Your downspout ends and transitions to schedule 20 PVC well before grade. I don't see what the issue would be pouring around it. You can wrap a thin layer of expanding joint membrane around it If that is your concern.
I don't see how moving the down spout around d the corner helps anything unless you dig up and re-run the drain pipe. I certainly wouldn't add a bunch of 90 degree fittings to bring it around the corner
Maybe I'm missing something?