This is actually an exceedingly complex problem and not one that you're likely going to be able to solve using conventional ArcGIS tools. To do this, you'll need to develop for each depression in your landscape the relation between depth and volume, i.e. the depth vs volume curve, which will be uniquely defined for each depression based on it's form. To calculate this curve you would essentially need to enter the height of each grid cell contained within a depression into a sorted list. Then visit each entry, from lowest to highest. For each new entry i of height zi calculate the sum of (zi - zj) x g2, where j is a grid cell within the depression of lower elevation than zi. This is an estimate of the volume of the depressional feature at a depth d = zi - z0 where z0 is the lowest elevation in the feature.
Now, since you've mentioned that you've defined the flowpaths and watersheds of each depression I assume that you plan on relating storm events which result in a specified depth of rainfall (p) over your defined watershed of area a, such that you can get a volume (p x a) and then use the volume-depth relation described above to fill the depression to a certain level, or depth. That's great and I've certainly seen this approach applied before, but you're missing an important complexity here (several actually). First, because a depression may well have several other upslope depressions within their watersheds, the watershed itself must be dynamically defined, e.g. if you have a storm of 15 mm then this upslope depression will become filled and its watershed will then contribute to and expand the watershed of this downslope depression. This quickly becomes an exceedingly complex thing to calculate although I have seen it done before. The next main thing that you may be overlooking is that the watershed areas that we define using DEMs in GIS are not really hydrologically realistic, i.e. the entire watershed for a downslope point will rarely if ever contribute flow to that point at any given time. Instead the catchment area that drains to a point of interest in a catchment itself will expand and contract based on moisture conditions much like the dynamic surface-saturated area concept. It's possible of course to try to model this dynamic expansion/contraction of the contributing areas to each depression using a hydrological model like TOPMODEL, but this complexity combined with the fact that you have fill-and-spill of upslope depressions resulting in thresholds or step-changes will mean that you are likely well outside the scope of what can be accomplished within a traditional GIS like ArcGIS and likely will need to program a custom solution.
It's an exciting problem and one that I have spent a great deal of time thinking about in the past. I recall a researcher at the Centre for Hydrology, the University of Saskatchewan named Dr Kevin Shook was looking at something very similar to this. You may want to contact him directly. I wish you the best of luck in pursuing this interesting problem.