My daughter has a dollhouse and some of the accessories she has make my design OCD act up. This swing in particular frustrated both me and her. The swing basically sits on the base with nothing but gravity to keep it in place.
I used a 1/16" drill bit in my Dremel press (since I never have a chance to use the thing) hoping that the extra high RPM would make four nice clean holes in the small wooden base.
Using four of the smallest nails that I could find and two small strips of plastic (the kind used in a blister pack) I made a some U-bolt like fasteners to keep the swing on the base.
I cut the strips to size then used the holes you drilled to punch through the plastic at the right spacing. Since the nails I had were slightly smaller than the holes I drilled I put a small bead of hot glue in each hole to make the nails stick. Placed the beam across the base and and nailed the strips in. Works much netter now!