I think you are on the wrong side of that retaining wall. You want to intercept the water in the soil, before it reaches any structure that water will corrode or erode. Drain pipes go BEHIND the wall on the side with the surcharge, the soil that that wall or foundation is holding back. They are also always lower than the footing of a foundation or wall or stair system. You want to use a perforated 4" plastic pipe below the footing (not on top of the footing) on the up hill side. Water flows through the soil to collect in this pipe to be drained away.
Dig a trench coming at us in the picture from the outside of the edge of your stairway, walkway. Looks like 4' from the foundation to the left and parallel to the home. Dig a trench as deep as the foundation's footing. If there is a basement this depth could be a good 6 to 8 feet deep and 2' wide. Install landscape fabric, a few inches of drain rock, install the perforated drain pipe wrapped in landscape fabric, the bottom of that pipe being LOWER than the foundation footing. Back fill with drain rock, use landscape fabric on top so that soil is unable to mix with the drain rock, top with soil or better, gravel.
Absolutely no water should be able to drain towards the concrete foundation. The perf pipe is always just below the lowest part of the footing of your foundation. That pipe has a definite slope towards your chosen drain outlet. That pipe should be at the very least 3' below the surface if you live with winters.
The second option would be if you find your foundation drain system is non existent or non functioning, I would pull all of that soil completely back from the foundation, re coat the asphalt on your foundation, allow to dry, install proper foundation drain system with drain rock, landscape fabric, more drain rock...either tying into the original system if it works at all or doing the entire foundation with fresh asphalt, foundation drain pipe system, landscape fabric. The subsurface soil has to slope away from your foundation. The perf pipe helps to collect any water that gets close to your foundation and diverts it away to either a sewer water system or a daylighted drain. Depends on your restrictions.
This is probably one of the most important aspects for the value of your home. Drainage. Without drainage the foundation will be short term. You'll get mold that could be very hazardous to some people's health. When considering any project that will need some heavy machinery the best advice is to get as much as possible done when the equipment is there the first time. I would ask for bids but chose the one who is able to see the entire project scope best such as calling for utility location, what to do with soil/debris, etc... not the one who is cheapest. Also, make dang sure they are licensed and insured have insurance of their own.
What I think you are trying to do is on the wrong side of that retaining wall. To amend your system those stairs get pulled out so you are down to the level of your footing. If you do not have a decent periphery foundation drainage system, I would consider redoing that entire system to do at this time. Best and cheapest in the long run. A good fix would be to just do this one side on the outside of that wall not the inside.
Procuring bids which also means professional advice is free and very valuable even if you decided to DIY.
You are thinking about day lighting your foundation drainage that is near a stream to run into that stream? That is a major no no with the rules in this country. Where exactly is it you live? All rivers, creeks even intermittent creeks have buffer zones that no one owns in this country. It is a very good thing but those who have riparian systems on their property usually learn that they truly don't OWN that part of their property. Please send more information. Giving answers about drainage based on little information is frankly irresponsible. Of ME. Please help by adding any information you can think of to help us be more responsible.