Framed[
 Text[Row[{"v = " , NumberForm[10.38, {4, 2}], " meters/second"}]]]

Use the following instead:

Print[Framed[
  Row[{"v = ", 
    NumberForm[Quantity[10.38, ("Meters")/("Seconds")], 2]}]]]

The issue you were having was because "v=" NumberForm[...] means Times["v=",NumberForm[...]] and so the syntax form of the underlying quantity was taking over, because a string times a quantity doesn't evaluate. Another example of this would be:

"v=" RawBoxes@InterpretationBox[10, 10, SyntaxForm -> Mod]

Thanks. Your solution is almost perfect, except by the lost of the output format.

Did you miss the part "Also Text is not needed though to my eye it is useful for formatting"? It's there for a reason..

If you are fastidious about the proper math formating (fonts and spacings), try this:

Print[Framed[ DisplayForm@ FormBox[RowBox[{"v", "=", NumberForm[Quantity[10.38, ("Meters")/("Seconds")], 2]}], TraditionalForm]]]

I found out about FormBox by hitting the menu Cell > Show Expression.