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#Mathematica, 71 bytes
Max[L/@Select[Array[a~IntegerDigits~#&,a=#,2],(L=Length)@Union@#==1&]]&
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#Mathematica, 7771 bytes
Max@Array[Length/@s[[#]]&,Length[s=SplitMax[L/@Array[IntegerDigits[a,#]&@Select[Array[a~IntegerDigits~#&,a=#,2]]]&2],(L=Length)@Union@#==1&]]&
#Mathematica, 77 bytes
Max@Array[Length/@s[[#]]&,Length[s=Split/@Array[IntegerDigits[a,#]&,a=#,2]]]&
