Algebras in a bicategory of spans

In general, it doesn't exist. That is, $\mathbf{Span}(\mathcal{E})$ doesn't have all EM-objects. If we embed $\mathbf{Span}(\mathcal{E})$ into the bicategory $\mathbf{Prof}(\mathcal{E})$ of internal categories and profunctors, then the EM-object of an internal category $C$ regarded as a monad in $\mathbf{Prof}(\mathcal{E})$ coincides with its Kleisli object, and is $C$ itself, regarded instead as an object of $\mathbf{Prof}(\mathcal{E})$. This makes it easy to see that most EM-objects in $\mathbf{Span}(\mathcal{E})$ do not exist: if $C$ had an EM-object in $\mathbf{Span}(\mathcal{E})$ then $C$ would be Morita equivalent to a discrete internal category.

In fact, $\mathbf{Prof}(\mathcal{E})$ is the "free cocompletion" of $\mathbf{Span}(\mathcal{E})$ under Kleisli objects (and hence also the free completion under EM-objects, since both are self-dual) in a suitable sense. The Lack-Street paper FTM2 that you mention is one important ingredient in this (it shows how to get the 2-cells right), but in order to get the 1-cells right one needs to look at a suitable kind of enriched cocompletion; see arXiv:1301.3191.