The set phrases очень даже and довольно даже convey the element of contradicting expectations, similar to the English word "too":
— Вы же ее не продаете. ― А вот и не угадали. Очень даже продаю // "But you're not selling it." "You guessed wrong. I am too selling it".
― Тебе, может, и не праздник, а я очень даже рад, что ты на свет появилась. // "It might not be a special day for you, but I'm actually (quite) glad you came into this world"
Ведь и он, как другие, постоял, постоял да и пошел ― и довольно даже свободно вошел… // But he, like the others, stood there for some time and then just went — and entered quite freely, too.
— Безобразенъ? — Нѣтъ, нельзя сказать, довольно даже интересенъ. // Is he ugly? No, you can't say that, he's actually quite personable.
The phrase довольно даже has fallen out of usage in the last hundred years, and additionally, Dostoyevsky reversed the word order. But it's the same set phrase, so you can't analyze it and translate just the word "даже" in isolation.
I would translate it this way:
Even though the family came to terms with the situation — and quite quickly, too,...