It is going to be very difficult (on a really good day) to color match that and smoothly blend it in along the edges so that nobody knows that there's been a "patch" there.
Matching stain is an artistic endeavor and will take time, patience and trial and error. Even if you have a can of the original stain with a sworn statement from the previous owner that this is exactly what they applied to the floor, the stain on the floor will have faded over time from what was originally applied, so a fresh application will still stand out. Additionally, the wood has been lightened by the cleaning process, though it hasn't been evenly lightened - there's still a dark spot in it and that will definitely telegraph through any sort of intentional staining process.
Ideally, you'd work on some scraps of wood from this project, apply various colors to find something that's similar, then if necessary, apply a second coat, or possibly a different color. Lather, rinse, repeat (keeping good notes on every sample) until you've found a satisfactory process that will get you as close as possible to the current color.
Since you don't have that luxury, you're going to have to work "live". I would suggest that you start with a very light color stain. You could apply it straight from the can as the manufacturer recommends, or you could pour some out into a pot and thin it with the appropriate thinner (water or turpentine/mineral spirits) and wipe it on, let it sit a few minutes, then wipe the excess off. Let it sit and dry for several hours, then look at it from several directions under natural light (looks like there's a window close by) and only artificial light (close the blinds or wait for night time).
You'll want to patiently repeat this process - if you get too much stain on too quickly, or start with too dark a color, you'll end up with this area being too dark, and your only options will be to live with it or to sand it off and start over. Again.
You'll have to eyeball the finish level to your satisfaction. It's possible that you may put one coat over the whole thing, then two or three coats between what you've sanded out and where the stain is. You'll also have to mentally adjust for the fact that the currently finished floor has a clear, glossy top coat of some sort on it, while your work area does not. This gloss coat will change the apparent color of the repaired area and will make it more difficult to accurately judge the final color.
You may have to apply a coat of stain, then immediately use a rag with some thinner on it (again, water or turpentine, depending on the type of stain you're using), wipe along the edges where it blends with the existing finish to "feather" the edges in to avoid a hard line between existing finish and new finish.
Also, since the stained spot is darker now, it's going to be darker once the finish has been applied. The good news, IMHO, is that it's somewhat subtle and it doesn't have a hard, defined edge. I think you'll probably just have to live with it and think of it as a natural discoloration in the wood. If anyone asks, call it "character". :)
Do note that it's possible that there has been no stain at all added to the wood and that the color you're seeing as a finish color is simply a few coats of a clear, gloss finish followed by the natural warming of the color of the wood as it's been exposed to light over the decades.
This may be a case of "do your best", then throw a rug over it or put a piece of furniture there to hide it so it doesn't bother you. Then it's up to the next owner to decide if it bothers them enough for a full refinish.
This is a DIY site and I strongly encourage and support the DIY effort. However, this may well be a case for hiring a professional. Personally, I do not have the artistic ability to judge colors all that well, nor do I have the experience to be able to mentally add a gloss top coat and know how that's going to change the apparent color of the finish. You might want to consider finding a couple of wood finish restorers (you might try a search for "antique restoration" or "wood preservation") in your area and ask them out to give a quote to do this repair. You may find that the price is painful, or it might be surprisingly reasonable, but in either case A) you'll know, and B) you'll have an experienced, artistic pro giving you an idea of what's involved. You can always turn them all down if it's too pricey and you decide to do it yourself.
Another option would be to sand down the whole floor and refinish the whole thing. Obviously, that makes this a much bigger task that will take more time, effort and money, but will make all but the stained spot itself disappear since the whole thing will be finished all at once.
This is the "nuclear" option and would probably be the last resort if you simply cannot get the repaired area to "disappear" to your satisfaction.
