The break is under the patio, but $5,500 was them going under the patio and staying within the confines of the driveway/sidewalk, so no concrete work. He put flags at the first 90 and where the cast gets better and he wrote the depth he was getting. That's where the 2'-4" and 2'-10" came from.
Considering the shallow depth and short run, it almost seems do-able with a shovel. We just planted a little tree in that bed which I would want to try and transplant to at least try and save it. Right there, a good bit of your digging is done. The place up the street from work rents a mini-excavator for $180/day.
In the picture I did, the top of the concrete patio is lines up with the top of the concrete foundation where it butts up to the side of the garage so ideally, the patio should be pinned to the concrete along the back and side. I figured the biggest problem would be digging it out so there was enough room to get under there and make the actual connection. The whole excavated area would be pea gravel or sand under the new pipe anyway, what if I went back in with sand or pea gravel under the patio?