It is hard for me to tell exactly what is where from your pictures, but if you have water running into the room then just putting in a drainage pipe is likely not going to fix the issue.
The best course of action is to properly slope the surrounding area. It should be below the threshold of the door, and then pitch away in all directions. If that cannot be achieved, then you need to put a drain in the area and run drainage pipe, at the correct slope, to daylight.
You could in theory use crushed stone (not chips, 3/4" clean) as a basin to collect the water, but given how much mud it looks like enters the mud room, that is going to silt up quickly, and then it is a bear to clean out.
A better option is to install something like this - NDS 12 in. x 12 in. Drainage Catch Basin, 2 Opening Kit 1200BKIT - The Home Depot
in the area and slope the surrounding area to the drain. Use the pipe with the foam peanuts around it closest to your mud room and the drain. Then use the rest of your perf or slotted pipe (whatever you have) for the rest of the run. The drainage pipe needs to be at a slope of 1/8" per foot (or better, if you have heavy runoff, 1/4" per foot). Holes on perf pipe go on the top and bottom, not the sides.
Bed the pipe in more 3/4" clean stone. If you want to get fancy you can line the trench with a geotextile before you backfill. No need for stone dust, in fact you do not want the stone dust, it will infiltrate the drainage structure.
You are also going to need a trench that is much bigger than 6"-8". You need room for the stone, and you need room to work to get your slope correctly.
Edit: I reread what you are planning to do. Creating a swale leading away from the door to an other spot by the shed should also work, as long as that is not where you plan to walk on a regular basis.
I would still put a catch basin at the end of the swale at your shed.
The best course of action is to properly slope the surrounding area. It should be below the threshold of the door, and then pitch away in all directions. If that cannot be achieved, then you need to put a drain in the area and run drainage pipe, at the correct slope, to daylight.
You could in theory use crushed stone (not chips, 3/4" clean) as a basin to collect the water, but given how much mud it looks like enters the mud room, that is going to silt up quickly, and then it is a bear to clean out.
A better option is to install something like this - NDS 12 in. x 12 in. Drainage Catch Basin, 2 Opening Kit 1200BKIT - The Home Depot
in the area and slope the surrounding area to the drain. Use the pipe with the foam peanuts around it closest to your mud room and the drain. Then use the rest of your perf or slotted pipe (whatever you have) for the rest of the run. The drainage pipe needs to be at a slope of 1/8" per foot (or better, if you have heavy runoff, 1/4" per foot). Holes on perf pipe go on the top and bottom, not the sides.
Bed the pipe in more 3/4" clean stone. If you want to get fancy you can line the trench with a geotextile before you backfill. No need for stone dust, in fact you do not want the stone dust, it will infiltrate the drainage structure.
You are also going to need a trench that is much bigger than 6"-8". You need room for the stone, and you need room to work to get your slope correctly.
Edit: I reread what you are planning to do. Creating a swale leading away from the door to an other spot by the shed should also work, as long as that is not where you plan to walk on a regular basis.
I would still put a catch basin at the end of the swale at your shed.
well that was an interesting find! So that has been going on for a LONG time and the former owners just ignored it 
Thank you. I do a lot by myself. Always have.
I did start working on it at 1800 til 1930. I'm brainstorming how to do the portion under the big gate. I think I'll take the grass off the top and but the flat stone up against the other grass and make it level after I back fill the drainage pipe. I have enough flat rock to make a little 1ft retaining wall to go around the fence and make a step up going into their yard. I'm only going to dog our to the post of the big gate. I'll get up early and start working on it tomorrow morning. It's my last day off and I want to swim too. My trusty shovel. I've had that girl for almost 10 years. We've dug a lot of ditches together. LOL
