I'm not going to answer the question directly - because to do so would require far more knowledge about your world than I care to get.
However, I am going to answer it indirectly - referencing the current US Navy and the Royal Navy at the height of the Empire.
To preface this - the assumption that because there is a single governing entity there is no need for an expansive Navy is IMO flawed. Even in peace time, a Blue Water Navy has 3 functions:
1: Showing the Flag - reminding everyone to play nice so that the big stick doesn't come out.
2: Humanitarian efforts - sometimes things go bad and a Warship is a great way to supplement any relief with Sailors, Supplies, Medical service etc.
3: Search and Rescue - although kinda similar to point 2, this is more to do with single incidents - like a Plane crash or similar.
Let's start with Capital Ships:
The US Navy currently has 9 Carrier Strike Groups, almost 2 for every Ocean.
At the peak of the Royal Navy - it had around 15 Battleships and Battlecruisers - around 3 per Ocean.
One in-theatre, one in reserve. The RN had Battleships and Battlecruisers counted in the same metric, but they aren't the same - so I'm using the one in, one out for brevity.
In your world, there would be sections of space that are divided up into Zones - most likely separated by the approximate distance it would take to traverse these Zones.
Therefore with your Capital ships, the smallest I would go for is 1 per zone with a percentage in reserve. e.g. say you had 10 zones, Maybe 17 Capital ships - one per zone and a reserve of 2/3rds (rounded up).
Next up would be your primary support ships - equivalent to a Cruiser - In modern times these are getting rarer as it's often more efficient to have a larger number of Destroyers, especially since a modern destroyer is capable of covering the roles traditionally associated with a cruiser.
But I like Cruisers so I'm going to include them, 2 cruisers per capital ship would seem a good number - this gives flexibility - if on patrol something has happened where sending the Capital ship is too much, then dispatching a Cruiser and a Gaggle of Destroyers makes sense, whilst retaining one with the Capital ship seems reasonable.
Destroyers! Mo Destroyers, Mo Navy.
These are the Work-horses of your Fleet, not only will they protect a Capital formation, but they are big enough and bad enough to handle most solo-missions. At time of writing, the US Navy has around 70 Destroyers, in a typical carrier group there are 4. 4 x 9 = 36, so we can presume that you want double the number you assign to a Capital formation.
Submarines and equivalent - in Space, the concept of a Submarine doesn't really work, so we are going to ignore these for now.
Logistics ships. Not very sexy, but if you can't get what you need to where you need it, you aren't going to be doing much.
Currently, there are 130 Non-Commissioned vessels in the USN fleet - but considering this is Space (Space is inherently hostile and so the need for fast logistics is increased) - I think you should have 20 times the number of Logistics ships as you do Capital ships.
Landing Ships/Assault Ships - most likely your larger ships will not be able to enter an Atmosphere, so you'll need some specialized ships to create inroads into a planet. Currently these are about a 3-to-1 for Capital ships - there's around 30 landing ships in the US inventory, but as above with Logistics, Planets and space is big, so you may want to increase that number.
I'm gonna be lazy here and use the US Coast Guard for this - they have 1,400 boats - again, cause space - you are probably going to want a very large number here, most likely dependent on how many planets you have - these vessels are small, cheap and not designed for inter-planetary operations - things like stopping smugglers etc. If we use the number of GPS Satellites in Orbit (a good metric as several have to be observable at any time to work) - 31 vessels of this small type per planet seems reasonable.
So to recap:
Capital Ships: 2 per 'region' of your world
Cruisers: 2 per Capital Ship
Destroyers: 4 per Capital ship, then double that for solo-missions
Logistics: Space isn't nice - 20 per Capital ship
Landing Craft: 3-6 per Capital Ship
Frigate/Littoral Craft: 31 per planet.
That's a Navy that can show the flag if needed, help in a Humanitarian crisis and do SAR without being stretched too thin.