Körner is referring to the Refutation of Idealism argument (B274–279), directed against the skepticism about the external world attributed to Descartes and Berkeley. The idealism in question is the "dogmatic" idealism concerning the empirical, hence the "empirical realism". The choice of words is unfortunate, however, since "empirical realism" is also one of the two names given by Kant to his own philosophy. The other one came to be much better known, and became its canonical label, the transcendental idealism. But Kant considered "empirical realism" to be equally valid, except that his meaning for it has little to do with what the term means today, which is closer to what is argued for in the Refutation, see Palmquist's Two Perspectives on the Object of Knowledge.
Here is Dicker's reconstruction of the Refutation argument:
1) I am conscious of my own existence in time; that is, I am aware, and can be aware, that I have experiences that occur in a specific temporal order. (premise)
2) I can be aware of having experiences that occur in a specific temporal order only if I perceive something permanent by reference to which I can determine their temporal order. (premise)
3) No conscious state of my own can serve as the permanent entity by reference to which I can determine the temporal order of my experiences. (premise)
4) Time itself cannot serve as this permanent entity by reference to which I can determine the temporal order of my experiences. (premise)
5) If (2), (3), and (4), are true, then I can be aware of having experiences that occur in a specific temporal order only if I perceive persisting objects in space outside me by reference to which I can determine the temporal order of my experiences. (premise)
6) Therefore, I perceive persisting objects in space outside me by reference to which I can determine the temporal order of my experiences. (1–5)
Permanence is required as a baseline for establishing temporal order among past experiences. Conscious states can not play this role, "this permanent something cannot be something in me, for the very reason that my existence in time is itself determined by this permanent something". The empirical "soul", or self, is a collection of appearances arranged in time, the metaphysical soul, along with the immateriality arguments, is dispatched in the Second Paralogism, see What are the problems with the argument for the mind-body dualism from immateriality of thoughts? The noumenal self is moot to the kind of "metaphysical" idealism that Kant is dealing with here. The entire argument concerns the realm of appearances, in space and time, we are talking about realism vs idealism about appearances, the noumena are out of the picture. Time itself can not supply the permanence either, for "time by itself is not perceived... Hence it follows that consciousness in time is necessarily connected also with the existence of things without me".
There are three common objections. One might suggest that the memory is unreliable, and hence reject premise 1). One can argue, contra premise 3), that conscious states (empirical ones, we are not talking about the "soul" here) can function as the permanent baseline in lieu of the external things. Kant could reply, I suppose, that this is highly implausible due to general obscurity of introspection (and Wittgenstein would agree). But, on Berkeley's view, the "external things", including clocks, are subject's mental states too, esse est percipi. So Kant's refutation falls short of its intended target. Finally, the "permanence", if established at all, is only of relative quality, flashing mental time stamps would suffice for establishing the temporal order too. See SEP discussion for more details. A recent defense of the argument is in Refutation of Idealism and the Distinction between Phenomena and Noumena by Edmundts in the Cambridge Companion to Kant's Critique of Pure Reason.