In my paper Condensations in higher categories joint with Davide Gaiotto, we claim the following answer. It deserves someone to write a careful model-dependent paper confirming it. I believe Martin Szyld has been thinking about how to do this.
Define a 2-idempotent to be an endomorphism $e : X \to X$ together with a retract $\mu: e^2 \Leftrightarrow e : \mu^*$, by which I mean 2-morphisms $\mu$ and $\mu^*$ such that $\mu \cdot \mu^* = \mathrm{id}_e$, which is associative, coassiciative, and Frobenius. [In spite of the name "$\mu^*$", it is just another 2-morphism, and isn't required to be the adjoint or dual in any sense beyond the requirement $\mu \cdot \mu^* = \mathrm{id}_e$. In particular, $\mu$ constrains but does not determine $\mu^*$.]
By Frobenius I mean that the three natural maps $e^2 \Rightarrow e^2$, namely $\mu^* \cdot \mu$, $(\mu \circ \mathrm{id}_e) \cdot \mathrm{assoc} \cdot (\mathrm{id}_e \circ \mu^*)$, and $(\mathrm{id}_e \circ \mu) \cdot \mathrm{assoc}^{-1} \cdot (\mu^* \circ \mathrm{id}_e)$, are all equal. [Actually, the Frobenius axiom together with $\mu \cdot \mu^* = \mathrm{id}_e$ imply the associativity of $\mu$ and coassociativity of $\mu^*$.] Here my notation is that $\circ$ is the composition in the 1-morphism direction, and $\cdot$ for 2-morphisms, and $\mathrm{assoc} : e \circ e^2 \cong e^2 \circ e$ is the associator.
This same notion goes by many names. For instance, it is a nonunital separated monad, or a nonunital special Frobenius monad. It is almost but not quite the same as "separable monad" used by Douglas and Reutter. In our paper, we give it yet another name — "condensation monad". But Reutter and I have started saying "2-idempotent" when we talk to each other, and perhaps it is the best name.
A 2-idempotent $(X,e,\dots)$ splits when there is an object $Y$, 1-morphisms $f : X \leftrightarrows Y: f^*$ [again, in spite of the name, I don't require any duality/adjunction], and some 2-morphisms and equations which I will list. First, I require the data of a retract $\phi : f\circ f^* \Leftrightarrow \mathrm{id}_Y : \phi^*$, i.e. 2-morphisms such that $\phi \cdot \phi^* = \mathrm{id}_{\mathrm{id}_Y}$. Now, using the retract $(\phi,\phi^*)$, I claim that you can give the composition $f^*\circ f$ the data of a 2-idempotent. Specifically, you set $\mu : (f^* \circ f) \circ (f^* \circ f) \Rightarrow (f^* \circ f) \circ (f^* \circ f)$ to be what you get by using an assotiator $(f^* \circ f) \circ (f^* \circ f) \cong f^* \circ (f \circ f^*) \circ f$ and then applying $\mathrm{id}_{f^*} \circ \phi \circ \mathrm{id}_f$ and then applying some unitors; and $\mu^*$ is the reverse. The last datum needed to say that $(X,e,\dots)$ splits is an isomorphism $e \cong f^* \circ f$ of 2-idempotents on $X$.
To connect with things you know: 2-idempotents are a version of monads, and splittings are a version of Eilenberg–Moore objects. The difference is that mine are not unital and are separable, in fact separated.
Then our claim is that a weak 2-category is Cauchy complete when (and only when) it is locally idempotent complete [i.e. all hom-categories are idempotent complete] and also every 2-idempotent splits. If your 2-category is locally idempotent, then a splitting of a 2-idempotent is unique up to unique isomorphism. I forget if this is true without local idempotent completion.
We also claim an explicit construction of the 2- (indeed, $n$-)categorical Karoubi completion. Given any 2-category, the first step is to locally Karoubi complete it. [Easy exercise: write down the 1-morphism composition in the local Karoubi-completion of a 2-category.] Now given a locally Karoubi complete 2-category $\mathcal{C}$, I build a new 2-category whose objects are the 2-idempotents in $\mathcal{C}$. A morphism $(X,e,\dots) \to (Y,f,\dots)$ is a morphism $m : X \to Y$ together with retracts $m\circ e \Leftrightarrow m$ and $f \circ m \Leftrightarrow m$ such that a bunch of equations hold making $m$ into a bimodule and bicomodule and cetera. A "bi-bimodule", perhaps? The 2-morphisms are natural: they are the homomorphisms of these bi-bimodules.
The interesting thing is the composition. Given $(m, \dots) : (X,e,\dots) \to (Y,f,\dots)$ and $(n, \dots) : (Y,f,\dots) \to (Z,g,\dots)$, I can look at $n \circ m : X \to Z$. Now I claim you can write down an idempotent $n \circ m \Rightarrow n\circ m$. The trick is to map $n \circ m \Rightarrow n \circ f \circ m \Rightarrow n \circ m$ where you use the $f$-coaction on $m$, and then the $f$-action on $n$. Well, you could have used the $f$-coaction on $n$ and then the $f$-action on $m$, but it turns out you will get the same thing. Since your 2-category is by assumption locally idempotent complete, you can split this idempotent. Actually, $n \circ m$ was already a bi-bimodule, and the idempotent you write down on it is a moprhism of bi-bimodules. So the splitting is a bi-bimodule. This is the composition in the 2-Karoubi completion.
You can see that to write this down as a bicategory, say, would require making some arbitrary choices: I just told you "split the thing", and not which splitting to take, so even if you started life as a strict 2-category, you won't end up strict, and you should not expect our construction to lead to any statement of the form "this is the Cauchy completion" in the world of strict 2-categories and their functors.
