Short answer: 'a deal' = 'somewhat' because 'a great deal' = 'a lot'. The sentence in question would feel much more natural to a modern reader if it had said:
... you thought we were a great deal happier than the King children...
Longer answer: I also trip over that phrasing: "...we were a deal happier than the King's children".
In slightly informal English, 'deal' can be used in a number of non-literal or idiomatic ways (literally it means 'a transaction or pact').
- It is a common intensifier to say "I am a great deal annoyed by that bird's cackling".
- "What's the deal?" means "What is going on?".
- It can be used to mean 'the matter at hand': "Hey, it's no big deal that you borrowed my fascinating book on anti-gravity; I bet you couldn't put it down."
These are mostly fixed phrases and can't be expanded very well beyond. I suppose at one time, as in the mid 1800's, 'that is a deal bigger' meant 'that is a lot bigger' (itself a bit of idiom), and the more intense version of 'That is a great deal bigger' took over and the weaker version hasn't been heard of since.