Normally, a small amount of free light chains are everywhere - about a 2:3 ratio of free kappa chains to lambda chains. If you were doing an immunoassay, it would count whole antibodies as well as free chains, and the ratio would be more like 2:1.
In myeloma, the production is monoclonal, so you will see a profile that deviates from this norm. It doesn't matter how much protein you have - indeed, if the kidneys haven't been damaged enough yet, the urine protein level could be normal - but the ratio will be altered because so much of one kind of light chain is being produced. Once there is sufficient glomerular damage, the Bence Jones proteins (crystallized agglomerations of light chain) will appear in the urine, as well as casts. Kappa is the most common, according to Goljan (p. 247 RR Pathology 3e).