Not every amalgam of solvable groups is embeddable into a solvable group; in fact, not even every amalgam of nil-2 groups is embeddable into a solvable group. For an example, see On certain embeddability criteria for group amalgams by B.H. Neumann and James Wiegold, Pub. Math. Debrecen 9 (1962) 57-64. The same paper establishes that CIM conditions will not suffice to characterize embeddability here.

   I am currently unaware of any general internal condition for embeddability which is both necessary and sufficient. There has been some work done placing restrictions on H, or on how H sits inside G and K (e.g. if it is normal).

   There is one general necessary and sufficient condition known, but unfortunately it is an external condition, invoking the existence of certain undetermined groups and certain maps from our amalgam to those groups satisfying sundry conditions. The result is due to Felix Leinen, as mentioned before. The paper is An amalgamation theorem for solvable groups, F. Leinen, Canadian Math. Bull. 30 (1987) no. 1, 9-18. The theorem also gives a bound for the solvability length of the embedding group in terms of the length of the original groups and the length of some of the groups which occur in the conditions.

   Since it involves external conditions, Leinen's Theorem seems very hard to apply to specific cases. I've been trying to see if it can be simplified by restricting the groups G and K to some subvariety, or in the case of special amalgams.