UTM is surely not the right projection. UTM 37 is centered on 39°E, and has a false Easting of 500000 metres to that. South of the equator, UTM 37S is measuring from the south pole to avoid negative values.
Apart from UTM, there was a Cassini-Soldner projection (possibly in feet instead of metres), and a 5 degree wide "East Africa War" transverse mercator CRS used in earlier times.
Using a Cassini-Soldner projection centered on 37°E/0°N with ARC 1960 datum, I get for your point in WGS84:
38.046282° East 0.059779° South (decimal degrees)
located about 43km east of Nkubu town.
Based on the documents linked in Convert Cassini-Soldner Projection coordinates from Kenya to UTM I suggest to use the following proj.4 parameters for a custom CRS:
For the central meridian 37° (valid 36°-38°E)
+proj=cass +lat_0=0 +lon_0=37 +x_0=0 +y_0=0 +a=6378350.87 +rf=294.26 +towgs84=-306,-6,-101,0,0,0,0 +units=m +no_defs
For the central meridian 39° (valid 38°-40°E)
+proj=cass +lat_0=0 +lon_0=39 +x_0=0 +y_0=0 +a=6378350.87 +rf=294.26 +towgs84=-321,0,-135,0,0,0,0
Ellipsoid parameters are taken from Clifford Mugniers article, Datum shift parameters were computed from the coordinate tables in the document by Gordon Okumu Wayumba