Maybe worth getting a second opinion from another mudjacker. That sidewalk looks like it is in pretty good shape.
Long term, you may have a grading problem because there is definitely a slope going into the house. Jacking the sidewalk slabs so they are pushing water away from the house will help a bit, but the problem is more likely in the sub-grade. You would need to remove the blocks and excavate down to the clay fill (remove all the topsoil). Topsoil does almost nothing to push water away from the house.
Then throw down clay fill, tamp it with a jumping jack (getting the slope right). For the side of the house like that, there is really no reason to have any topsoil on the side of the house at all. It is just there to make it easier for weeds to grow. Might as well go with a weed mat and some rock to fix the problem permanently.
If you go to sidewalk steppers (24" x 24") and install them properly using tamped 3/4" crushed road crush and no sand, you can save a lot of money. If they are installed with EdgePro and 6" spikes they will stay in place quite well over time. A concrete diamond blade on a skillsaw ($60-$70) will easily cut through the stones to get the angles right you need for the sidewalk.