First Determination, Who Built The Wall ?
actual facts may not be available, but the probability can be determined.
If the street was built before the house, then it is very likely that the house builder also built the wall.
If the home was built before the street, then it is possible the city built the wall. However the homeowner at the time may have built the wall, to provide support. Despite the fact the homeowner built it if it was for reasons of maintaining support the city is liable to repair it.
If the wall is not required for support then the city should just remove it and slope the bank to the property line. If it can be shown that original grade was raised on the property the city may also remove that fill if it is unsafe.
It is hard to determine from the single photo if the wall indeed supports the foundation. My best guess is that the wall was constructed to provide yard area, not lateral support, but a general investigation of the street and adjacent lots could give a more comprehensive view of the initial situation.
I Googled the street address and looked at the site in several views. The Street appears to follow the general terrain. The wall appears to have been built to provide a level yard. If the homeowner does not want to pay for repair the city should remove the wall, after notice that any future wall at that location must be on private property. If she refuses again it is her loss of useable real estate.
Paul in PA, PE, PLS
I suspect the City is correct; it is her responsibility to maintain the retaining wall because it is a street easement, not a fee adjoiner.
If that wall comes down and kills someone then the few thousand dollars required to fix it will seem like pennies in comparison.
We had a similar situation when I worked for State Parks only the wall was holding up the street. Our local District management was arguing with the County Engineer about who should pay to fix the problem (no one has the budget of course). I told them they are being stupid; we are two government agencies we should be trying to find a solution to the problem instead of pointing fingers. All we need is that wall to come down and badly hurt or kill someone and both agencies will look bad, really bad. Naturally they didn't listen to the stupid Land Surveyor.