If events unfolded as you say (and Japan Airlines says they did not, see below), I'd say you are owed about $2,100. That's the approximate difference between the $3,400 you paid for your replacement trip and the value of those 100,000 refunded Alaska miles, which, according to the personal finance site NerdWallet, is about $1,300.
When airlines cancel flights, they are generally obliged to offer passengers a choice between their money back or rebooking. But the devil is in the details: A refund is rarely enough to purchase a new last-minute flight, and rebooking is a slippery concept whose definition depends on whichever country's (or countries') laws govern the flight, as well as the specific airline's terms and conditions.
In your situation, the details were in your favor: JAL's contract of carriage does say the airline can book you on a competitor, presumably if it doesn't have anything available.
Collection
[
|
...
]