An algorithm is presented for the construction of fixed-length insertion/deletion correcting runlength-limited (RLL) codes. In this construction binary (d,k)-constrained codewords are generated by codewords of a q-ary Lee metric based code. It is shown that this new construction always yields better codes than known constructions. The use of a q-ary Lee (1987) metric code (q odd) is based on the assumption that an error (insertion, deletion, or peak-shift) has maximal size (q-1)/2. It is shown that a decoding algorithm for the Lee metric code can be extended so that it can also be applied to insertion/deletion correcting RLL codes. Furthermore, such an extended algorithm can also correct some error patterns containing errors of size more than (q-1)/2. As a consequence, if s denotes the maximal size of an error, then in some cases the alphabet size of the generating code can be s+1 instead of 2·s+1.