How can I convert a string to a float in mysql?

It turns out I was just missing DECIMAL on the CAST() description:

DECIMAL[(M[,D])]

Converts a value to DECIMAL data type. The optional arguments M and D specify the precision (M specifies the total number of digits) and the scale (D specifies the number of digits after the decimal point) of the decimal value. The default precision is two digits after the decimal point.

Thus, the following query worked:

UPDATE table SET
latitude = CAST(old_latitude AS DECIMAL(10,6)),
longitude = CAST(old_longitude AS DECIMAL(10,6));