I mistakenly booked two flights on the same day in and out of different Tokyo airports. I fly into Narita from Seoul and then out of Haneda. I'm wondering if this amount of time is feasible. Any experience would be appreciated.

ICN to Narita - land at 12:50pm

depart Haneda at 5:15pm

Do we have enough time (assuming everything is on time) to land at 12:50pm at Narita (exit plane, luggage, customs, etc.) and take the train to Haneda (arrive, check-in, security) and make our flight to Chicago.

How annoying! This is a math question. Is 4 1/2 hours sufficient.
Known factors are: Arrival at Narita: 12:50. Train (or bus) time to Haneda: Approx 1 1/2 hours. Train departures from Narita: Approx every 40 minutes.
Unknown factors are: Time getting off plane, getting luggage, going through customs; check-in, security, time from security to gate.
The unknown factors may be less time if you are traveling business/first class.

So, assuming 2 hours 10 minutes max (taking train and possibly just missing the train and having to wait for the next one) in transit time, airport to airport, that leaves you 2 hours 5 minutes for the unknown factors time.

The time it takes to clear immigration is the key issue for you. On our last trip in November it was over an hour in a snaking line. My cousin just returned last month and said it took almost 3 hours, yikes!

I would not like that connection as it’s hard to tell how long to get go through immigration and customs and get your bags assuming you checked some. Also assuming your flight from ICN is on time. Is this all on one ticket? Train takes about 2 hours, bus 70-90 minutes depending on traffic. Private limo would be faster as you would not have to wait for it. You then have to go through immigration and security at Haneda and get to your gate probably within 15 minutes of departure. Lots of potential road blocks.

Day of week will also affect transfer times if using bus/car. What happens to you if you miss your flight home?

Definitely use this:
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Are the tickets on the same airline? If so, you should be covered if you miss your connection, since they sold this connection to you. Day of week will make a difference. I would not use a taxi. It could take a lot longer than the train.

From Seoul to Haneda air fare is about $100. Maybe you are better off cancel the first leg and try another flight to Haneda. Narita to Haneda transfer is not cheap, around $25