The answer is yes:
Claim: Let $\mathcal K$ be a 2-category. Then $\mathcal K$ admits Eilenberg-Moore objects if and only if $\mathcal K$ admits terminal adjunctions inducing given monads.
If you think about the proof below, you'll see it also shows that this is true for a particular terminal adjunction / EM object, even if these constructions aren't assumed to exist in general
Proof: As you say, Street proves one direction, so we prove the other. Let $Mnd(\mathcal K)$ be defined as usual. Let $Adj(\mathcal K)$ be the 2-category of adjunctions in $\mathcal K$, where a 2-cell is a 2-cell between the right adjoints. We have forgetful 2-functors
$$ Mnd(\mathcal K) \leftarrow Adj(\mathcal K) \xleftarrow{const} \mathcal K $$
By hypothesis, the leftmost of these has a right adjoint. We wish to show that the composite (sending $K \in \mathcal K$ to the identity monad on $K$) has a right adjoint (this characterization of EM objects is of course also due to Street in the same paper). It will suffice to show that the rightmost 2-functor has a right adjoint. And it does -- the right adjoint sends $(L : C {}^\to_\leftarrow D : R) \mapsto D$.
(The funny thing here is that the diagram above doesn't seem to be induced in an obvious way by somehow "homming" some diagram into $\mathcal K$. At least $Mnd(\mathcal K)$ is the lax Gray internal hom $[[ Mnd, \mathcal K]]$ -- or equally the lax functor category $Fun^{lax}(1, \mathcal K)$ -- but I don't know how to construct $Adj(\mathcal K)$ except "by hand". As a result, the verifications in the above also need to be done "by hand". But it works out!)