What breaker size for box containing two GFCI circuits

You will also likely need to replace the cable with larger gauge wires, though without knowing what's there right now it's impossible to say for sure. Also, whenever you replace old electrical equipment, you are generally required to adhere to the latest code, so it's possible there are other changes you will need to make too. This might include AFCI breakers.

If you are unsure, always best to check with a licensed electrician.

The pertinent parts of Article 680 I have quoted here:

680.12 Maintenance Disconnecting Means One or more means to simultaneously disconnect all un-grounded conductors shall be provided for all utilization equipment other than lighting. Each means shall be readily accessible and within sight from its equipment and shall be located at least 1.5 m (5 ft) horizontally from the inside walls of a pool, spa, or hot tub unless separated from the open water by a permanently installed barrier that provides a 1.5 m (5 ft) reach path or greater. This horizontal distance is to be measured from the water’s edge along the shortest path required to reach the disconnect.

680 Part IV. Spas and Hot Tubs 680.40 General Electrical installations at spas and hot tubs shall comply with the provisions of Part I and Part IV of this article.

680.40 refers to Part I and the requirement for a disconnect is in Part I (630.12).

This commentary is from the 2011 NEC Handbook:

A readily accessible disconnecting means is required to be located within sight of pool, spa, and hot tub equipment in order to provide service personnel with the ability to safely disconnect power while servicing equipment such as motors, heaters, and control panels. Underwater luminaires are not subject to this requirement. The proximity of the disconnecting means to the pool must be not less than 5 ft unless the disconnecting means is separated from the water by a permanent barrier.

Notice here it says unless the disconnect is separated from the water by a "permanent barrier". If you have a permanent barrier such as a partition between the tub and the disconnect it can be closer than 5 feet. Otherwise 5' is your minimum distance.

So, you were reading the code correctly you need a disconnect in-sight of the spa and mounting next to the main panel will work just fine. Or using a breaker in the panel will work as you and Speedy Petey were discussing.