Was a building permit obtained for the addition, or was it "pirated"? Located in a fairly new subdivision, or out on a residential estate or whatever? Would the possible cause of the settling seem somewhat obvious if you knew what to look for, or is it not readily apparent?
It would be wise to determine what the problem is, before throwing solutions at it. It would be rare for it to be necessary to drill down to bedrock for a single-family house in Iowa, although in the southerly half of Iowa bedrock will be encountered by standard construction, NOT a great situation since lots of it is expansive shale.
Start with the City, and from there it might be wise to have the situation investigated by a professional soils engineer, someone like Allender Butzke,
http://abengineers.com .
According to your post you're located in Ankeny, and if it was permitted and inspections were requested, the City would have at least looked at the foundation excavation/pour. First and foremost, that it's below frost depth. One might think that frost can't move a house addition, but it can and will raise it in the winter and settle back down in the summer. It's not a straight up and down process, and the addition will gradually separate from the house to some degree. Expansive shale does the same thing, but via wetting and drying. I'd not think that you'd encounter shale in Ankeny, but MAYBE, not knowing exactly where your house is.
Secondly, there're assumptions made about typical soil bearing capacity, and there are "standard" foundations constructed for the typical soil conditions. If your conditions are atypical, the "standard" foundation design/construction wouldn't have been appropriate. If it's a new subdivision, everything within your lot probably was controlled grading BUT that my have only been done for the anticipated footprint of the house, and not for where the addition is located. That could result in different soil bearing capacities.
If the prior owner/their contractor just dumped fill to build up the location for the addition (hard to compact it properly next to a basement without caving in the basement), that fill is probably being compacted now by the weight of the addition, causing the addition to settle.
If your lot slopes, the area of the addition could be sliding due to the addition's weight, whether because they created a level spot for the addition via fill that's lot integral to the original soil, or because the foundation didn't go deep enough to be below the slope's "angle of repose".
Lots of Ankeny is wet, originally wetlands prior to farming, high organic content that could compact under the weight of the addition, maybe "dewatering" the soil will stop or at least slow the settling.
That's long enough to tell you that there are lots of potential causes for your apparent problem, and correspondingly lots of correct and incorrect solutions. Start with the City, then depending on what you find out, probably retain a professional soils engineer, so you only spend money once, on a real solution.