Python 2, 42 bytes

f=lambda k,n=0:n*(max(`n`)<'2')or f(k,n+k) 

Try it online!

Python 2, 42 bytes

f=lambda k,n=0:n*(max(`n`)<'2')or f(k,n+k) 

Try it online!

Full program, same length:

a=b=input() while'1'<max(`b`):b+=a print b 

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Python 2, 46 bytes

f=lambda k,n=0:max(`n`)<n*'1'and n or f(k,n+k) 

Try it online! fails for testcases 9 and 18 because of maximum recursion depth.

Python 2, 42 bytes

f=lambda k,n=0:n*(max(`n`)<'2')or f(k,n+k) 

Try it online!