I would use "I'm not doing so well" here instead; though I can't rally find fault with your version, it seems a bit "quaint" in my eyes (or maybe just BE?). In AE, one might actually say "I'm not so good" instead, which, however, can also mean that you're not so good at (doing) something, so it can be confusing.

There are other distinct subtleties with this type of phrase. "I'm well," for instance, would mean "mir geht es gut," whereas "I'm good" could also mean "ich habe genug" (for instance when asked if I want a second helping of food and I've had enough) or "mir geht es gut." As usual, only context will let you know what's meant, but you don't really have that problem with "I'm doing well / I'm not doing (so) well."

Ich habe dazu das folgende gefunden http://owl.english.purdue.edu/owl/resource/537/02/

es scheint als ob das Wort well in bestimmten Fällen sowohl as Adverb als auch als Adjektiv gesehen werden kann. Dies ist weiter unten auf der Seite unter 'Common errors' zu finden.

I am good hört sich für meine Ohren sehr AE an obwohl ich es auch schon vereinzelt in London gehört habe.

I've recently noticed in England that most young people will answer "I'm good" to the question "How are you? / How're you doing?" I think it sounds strange, having been brought up on "I'm fine. / I'm well." I think the fact we always used to use an adverb has to do with the point that the question is about how one is feeling - and you wouldn't normally answer a "how" question with an adjective.
Whatever the grammar of it all, I'm fine /I'm well is still standard BE, but I'm good seems to be acceptable with young people in informal situations.

'I'm not so well' isn't wrong, but it doesn't sound very idiomatic to me in normal conversation. Literally, 'well' did historically mean 'healthy' (= gesund), but this historical sense isn't used as often in modern English except in certain limited contexts, like the verb 'to get well' (= return to health after an illness), or phrases like 'He's not a well person' (= He's not in good health).

To express a temporary state of health or emotion, instead of just 'be' alone, we often use a verb like 'do' or 'feel':

(1) I'm not doing very well
in life - I'm depressed, discouraged, having problems

(2) I don't feel very good
inside myself - (a) physically: I'm worn out, my back hurts, I might be getting the flu; (b) mentally: I'm uncomfortable/dissatisfied (I don't feel very good about it = I'm uncomfortable/dissatisfied with it, e.g., a situation)

(3) I don't feel very well
old-fashioned, formal, or BE for (2)(a)


And of course you can use other tenses:

I haven't been doing very well lately.

I haven't been feeling very good/well lately.

and other adjectives:

—How's it going?
How're you feeling?

—Not very well. (standard/formal)
Not very/so good. / Not so well. (normal)
Not so great/hot. (casual)

Some of this may be more AE; BE may have other options.

'I'm good' can indeed now mean 'I'm fine' or 'Ich bin zufrieden,' even though literally and traditionally it means something more like 'I'm a good person' or 'Ich bin brav.' You could think of it as current slang in the sense that it's a fairly recent phenomenon and first became popular among younger generations.