zoning may not be the problem

The new unit was built in 1996 which is probably long after the zoning was already defined.  It is only 1 bedroom, so it is possible that it is an accessory dwelling unit (also called in-law unit or granny unit) which is allowed in single family residential zoning.  What you need to check is the permit for the new building with city planning/permit department. If there was permit approaved by city, then everything is ok.  if no permit was ever done, then check if accessory dwelling is allowed for your property/city, if allowed, get the permit to legalize the unit.