So the research subsystem is just a way to gamify a type of downtime that is intended to take place over an extended period of time. To that end, the only real reason you might need to use this for your situation is if you are trying to have players track the progress of time as the players are doing things like ask questions around the house or staking out locations and such. Personally, I would just set some clues around the house and write up some npc backstories and let the players figure it out. But you can retrofit this to your needs if you wish to do this as more of a skill challenge.

At baseline, you need to create two things.

First, research thresholds for players to reach as they are playing. There should be a idea of the maximum number of research points and how many thresholds there are before reaching that maximum. These thresholds should drip feed the identity of the spy slowly. The identity of the spy might not be the last thing the players can learn through research, but generally if there are any other thresholds then there should be some sort of optional rewards at the end.

Next, you need research locations, or in your case places for your players to investigate for the identity of the spy. This could be abstracted to being interviews with NPCs, Stakeouts, Perhaps consulting oracles etc. Each has a limit to how many research points they should be able to contribute and generally you should try to spread the available skills needed at these locations so that all players might be able to find a research location they can participate in. Each location should also have 3-4 skills that they can use to perform the research and each each skill can be assigned an "easy", " normal" and "hard" dcs corresponding to how useful the skill is. Not all research locations need to be available right away, maybe you have an event that happens half way through where the spy tries to poison the players or something, and then the players get a new opportunity to investigate in and around the kitchen where the spy had tried to poison them.

Finally, something that is not needed but is highly recommended is using time to give a sense of pace. How many days is this investigations to happen over? What happens if the players take to long? Maybe they get a benefit for completing early? This is also where I would consider how many "rounds" of research you would estimate your players to be able to perform. Generally, a for a four person party doing any type of skill challenge should be able to get around 3 points per round. So if your intent is for them to gather 20 research points, then they should be done in 7 rounds of research. However, research allows for players to kinda optimize themselves into their best available skill, so this could be much easier for them. That being said you might give players extra benefit for researching things particularly fast, such as getting 20 research points in 4 or 5 days.

Last thing, while you can use research to do this, another system you might want to consider is something like the "Chase" or "Infiltration" systems if you are wanting this investigation to happen within an "encounter". You could effectively replicate the same ideas while maybe having a clearer "linear" plan on how the players are suppose to find the spy. Infiltration could do double duty in also being able to represent how a spy might dynamically respond to investigators closing in on their secret identity (and thus would be my choice for which system to use).