Looking at your example of mouse, it seems you are going to use, and are asking about, the ‘old-fashioned’ fine-structural mouse, as used originally by Dodd and Jensen in their papers, and by Dodd in his book. It is crucial to say this since the ‘modern’ definition involved a reorganisation of these hierarchies following a suggestion of S. Baldwin in the 1980’s. The ‘modern’ organisation is the current approach to mice in all the various versions of Jensen and Mitchell-Steel. (It essentially ensures that a strong form of GCH applies level-by-level, which the Dodd-Jensen approach did not. This property is called (strong) acceptability.)
The upshot is that Dodd-Jensen have mice but a special subclass of these are the ’critical mice’ (and $O^\sharp$ is the first such, $O^{\sharp\sharp}$ is the second such, &c.). A critical mouse engenders an inner model by iterating it’s measure out through the ordinals. A non-critical DJ-mouse does not. (This is a major difference with modern mice: iterating a modern mouse by its topmost active extender repeatedly will always leave behind an inner model.)
So mouse from now on means a Dodd-Jensen mouse, and $K$ will be the Diodd-Jensen core model built below the first inner model with a measurable cardinal.
One may construct (non-critical) mice from critical mice, so that, as you say in $L[O^\sharp]$ then there are many mice definable from $O^\sharp$. You mention infinitely many but in fact there are a proper class many of such. So the length of the mouse order in $L[O^\sharp]$ is $On$ itself. In $L[O^{\sharp\sharp}]$ it is $On + On$ and so on.
More manageably: in any model or V, we may consider the mice in HC, or in $H_\kappa$ for some regular $\kappa$. These can all be iterated to a common hierarchy generated by the c.u.b. filter on $\kappa$, $F=F_\kappa$ say. This common iterate is of the form $J_\theta^F$ and is the direct limit then of all the mice in $H_\kappa$. The ‘strength’ of the mouse ordering can then be gauged by properties of $\theta$, (if it admissible, if it is . . .) but always $cf(\theta) = \kappa$ in any universe. (As an example, in $L[O^\sharp]$, thinking of the mice in $H_{\omega_1}$ they iterate to $J^F_{\omega_1 + \omega_1}$ where $F$ is the c.u.b. filter on $\omega_1$. In $L[O^{\sharp\sharp}]$, $\theta$ would be $\omega_1 \cdot 3$.)
Thus, to answer the question specifically, the mouse ordering in $L[O^\sharp]$ of the mice in $H_\kappa$ has length $\kappa$. In $L[O^{\sharp\sharp}]$ it is $\kappa\cdot 2$ &c. In the least model closed under $\sharp$’s it will have length $\kappa\cdot\kappa$. Once the model becomes even a little bit thicker there is no handy description of the length of the ordering in terms of some ordinal arithmetic. (Some of this is summarized in Sect. 3
of my “On unfoldable cardinals, $\omega$-closed cardinals and the beginnings of the inner model hierarchy”, Archive for Math. Logic, vol. 43, No. 4, (2004), pp.443-458.)
The common iterate $(J^F_\theta, \in F)$ for $F=F_\kappa, \theta = \theta(\kappa)$ Dodd and Jensen called the “$Q$-structure at $\kappa$” - thus $Q_\kappa$ (but again this has a totally different usage in modern mouse theory). Once the $Q$ structure is admissible interesting things can happen. As a sample (see again cited paper):
Theorem (V=K) If there exists a $\kappa$ with $Q_\kappa$ admissible, then
$\forall D \subseteq \omega_1( D$ is universally Baire $\leftrightarrow \exists r\subseteq \omega (D \in L[r])$.
Theorem (V=K) If $Q_{\omega_1}$ is admissible, then $\delta^1_2=\theta_{\omega_1}.$
These results are not lost by converting to the modern mice concepts. There are reasonably natural analogues. And to emphasise, the Dodd-Jensen $K$ has exactly the same sets as the 'modern' $K$ below a measurable cardinal. Most simply by forgetting about the non-critical mice in the Dodd-Jensen mouse ordering you will get an ordering of the same order type as the modern one of 'active' mice (or equivalently the order type of the inner models of $K$ under comparison).
