Repeated game with stage game has more than 1 NE Given the game like Prisoner's dilemma (infinitely undiscounted) and each person has 3 actions in every period: confess defect and kill, and the payoff matrix have 2 NE as (D,D),(K,K).

Can I still use the property that if in any stage NE is played, then that kind of strategy is SPNE.

It seems work for me to prove that any sequence like dd,kk,kk,kk,dd....randomly combination of dd,kk, is SPNE.

Is that correct? Can I generalize that property?
Though a bit unconventional to assume an infinitely repeated game without discounting, the short answer to your question is: yes. Any sequence of stage game Nash Equilibria is supportable as a SPNE. See, for example slide 11 of this deck, or these notes from Levin, starting on page 6. This is more directly addressed by the definition of SPNEs, found on page 137, Definition 132 of these notes. Or where those Columbia notes suggest on page 142: "For example if there are multiple Nash equilibria in any of the final subgames then there are multiple sub game perfect Nash equilibria in the game itself"

Though you should note that, given your problem's construction (with no discounting), the set of SPNEs grows substantially, as you can see.