Infinite dimensional 2-Hilbert spaces Is there a definition of an infinite dimensional 2-Hilbert space?
Finite dimensional 2-Hilbert spaces have been discussed by Baez in 
http://arxiv.org/abs/q-alg/9609018
In the more recent paper by Baez, Baratin, Freidel and Wise 
http://arxiv.org/abs/0812.4969
a notion of infinite dimensional 2-vector space is discussed, building on work by Crane, Sheppeard and Yetter. They also have a few proposals for what an infinite dimensional 2-Hilbert space should be, but "the details still need to be worked out". Is that the current state of knowledge?
Another way of asking this question is the following. If I want to see a field theory that has an infinite dimensional Hilbert space of states as an extended field theory, what kind of objects should I assign to codimension 2 manifolds?
 A: An answer to the question in the second paragraph (what kind of objects should a quantum field theory assign to codimension 2 manifolds?) can be found in my paper http://arxiv.org/abs/1304.7328v2.
I will argue that this also provides an answer to the OP's main question (what is a possibly infinite dimensional 2-Hilbert space?)

The answer is: it's a von Neumann algebra (or, possibly better, the module category of a von Neumann algebra).

Note that the kind of von Neumann algebras I have in mind are type $III$ factors and that for such an algebra, there is very little difference between the algebra and its representation category. If one excludes the zero module (and modules on non-separable Hilbert spaces), then the von Neumann algebra itself, viewed as a one object category, is equivalent (in the usual sense of equivalence of categories) to its representation category. In other words, such a von Neumann algebra has only one non-zero module up to isomorphism (excluding non-separable modules), and the endomorphism algebra of this unique module is the von Neumann algebra you started with. Note that this unique module is never irreducible: if you direct sum it with itself, you get again itself.
One can actually be more precise: the von Neumann algebras that show up in quantum field theory are always hyperfinite type $III_1$ factors (unless they are Morita equivalent to something finite dimensional). By a famous theorem of Connes and Haagerup, there is only one hyperfinite type $III_1$ factor (subject to appropriate separability assumptions) up to isomorphism. Thus, if one takes the point of view that an ``infinite dimensional 2-Hilbert space'' is the same thing as a hyperfinite type $III_1$ factor, then the Connes-Haagerup theorem says that there's a unique infinite dimensional 2-Hilbert space, up to isomorphism. This is the analog of the well known  and easy fact according to which there exists only one separable infinite dimensional Hilbert space, up to isomorphism.
Now one might ask: what is the 2-inner product on $M$-Mod? (here, $M$ is any von Neumann algebra).
For that, note that there is a complex-antilinear functor $M$-Mod $\to$ $M^{op}\!$-Mod given by sending a Hilbert space $H$ with action of $M$ to its complex conjugate $H\mapsto \overline H$. Here, the right action of $m\in M$ on $\overline H$ is given by $\overline\xi m:=\overline{m^*\xi}$. Given $H\in M$-Mod and $K\in M^{op}\!$-Mod, one can form the relative tensor product $K\boxtimes_M H$ (also explained here), which is some kind of fancy version of the tensor product over a ring. That's the 2-inner product. Namely, for two $M$-modules $H_1$ and $H_2$, we define $$\langle H_1,H_2\rangle := \overline{H_1}\boxtimes_MH_2 \in \text{Hilb}.$$
To contrast with Turion's answer, let me emphasise that a 2-Hilbert space is not enriched in Hilbert spaces. It is enriched in von Neumann algebras.
Also, a 2-Hilbert space doesn't have a product with $\text{Vect}$ for scalar multiplication. It has a product with $\text{Hilb}$.
As a final remark, let me point out that my ``2-Hilbert spaces'' are suitable for representing 2-Groups (topological groups with a $U(1)$-valued 3-cocycle in the sense of Segal), just like Hilbert spaces are good for representing centrally extended groups (topological groups with a $U(1)$-valued 2-cocycle).
A: This may be more of a comment or opinion.
A 2-Hilbert space is a Hilbert space on two levels: It's enriched in Hilbert spaces (morphisms) and it's a categorification of a Hilbert space in the sense of having biproducts for addition, Deligne-product with $\mathrm{Vect}$ for scalar multiplication and dual structure for the sesquilinear form. Which of those levels do you want to have infinite dimensional? The first one shouldn't be a big deal, you just allow for infinite dimensional Hilbert spaces as $\mathrm{hom}$-spaces.
The second one is the harder one. I personally think that it is no big deal "formally", it just amounts to dropping the requirement that the 2-Hilbert space be finitely semisimple. The question is, will you find interesting examples that are actually any use? You have the same in vector spaces. I can cook up tons of infinite dimensional vector spaces, but how do they become interesting and tractable at the same time? By adding some extra structure, say as a Banach space, a $C^*$-algebra, an $\mathrm{L}^2$-space over some measurable space or so. In that sense, I'd say that there already is a theory of "dull" infinite dimensional 2-Hilbert spaces, but people are thinking about interesting ones with extra structure, such as having a measure on objects.
As for your final question, I'd guess it will depend on the context of what you're doing. Are you considering some $n$-dimensional moduli space ($0 < n < \infty$)? Smooth groupoids? Then the approach with measurable categories may suffice. If your additional structure is different, a different variant might be needed.
