FullSimplify works, but it is much slower:
Timing[mkAlgebraic[thing]] (* {0.052406, Root[1 - 10 #1 + 5 #1^3 &, 2] + Sin[1]} *) Timing[FullSimplify[Re[thing]] + FullSimplify[Im[thing]]] (* {3.76156, Root[1 - 10 #1 + 5 #1^3 &, 2] + Sin[1]} *)
How about something like the following?
FullSimplify[Re[thing]] % // N FullSimplify[Im[thing]]
(* Root[1 - 10 #1 + 5 #1^3 &, 2] + Sin[1] *)
(* 0.941979 *)
(* 0 *)
