From what I've read, Sargent paints in one layer. The method is not complicated. And yet, knowing that, painting like Sargent is still really hard.
Have respect for many (not necessarily all :)) of the artists you see in museums. It's serious skill. But yet, do not just revere blindly. It is possible to learn what they know and do what they do.
Also, don't disregard people who are still alive. With the internet and so many books and libraries out there, it's very easy to learn whatever skill one is interested in. Contemporary people can be just as good as the best in the past.
When I was copying this one, I think I had a lot of fun. I think my colors came out a lot more grey than they should be. Also it's not just the photograph, but in my actually painting everything seems blurry. The paint didn't seem lively even though I remember putting a lot of energy into the marks. I think I did not use enough paint.