How to get Planck length in meters to 6 decimal places

You can use the relative uncertainty 1.1*10^-5 from the CODATA website you also referenced, and use Around to construct a value for the Planck length with uncertainty which has a much nicer formatting for showing the relevant digits of the number:

Around[UnitConvert[Quantity["PlanckLength"], "m"], Scaled[1.1*^-5]]

(1.616255±0.000018) x 10^-35 m

Since the relative uncertainty is also encoded in the precision of the number in Mathematica like Henrik pointed out, we can also use that to compute the relative uncertainty (the precision of the number representation in Mathematica not always coincides with the measurement uncertainty, but in this specific case it does):

Around[#, Scaled[10^-Precision[#]]] &[UnitConvert[Quantity["PlanckLength"], "m"]]

(1.61626±0.00004) x 10^-35 m

which is for some reason off by a factor of two. Correcting that factor gives the same result as earlier:

Around[#, Scaled[0.5*10^-Precision[#]]] &[UnitConvert[Quantity["PlanckLength"], "m"]]

(1.616255±0.000018) x 10^-35 m

The 4.652... at the end of 1.61625500000000006684132`4.652207380644164*^-35 should tell you that Mathematica knows this constant only up to 4.65 decimal digits.

You have to enforce first to treat the number as a higher precision number first:

NumberForm[
 SetPrecision[
  QuantityForm[UnitConvert[Quantity["PlanckLength"], "Meters"], 
   "LongForm"], 7],
 {7, 6}]

1.616255 * 10^-35 meters