～か　(who is the sentence directed to?)

Hello,

I need a little help with determining the meaning of a sentence in this video (動画の再生｜応用スキット｜第11課). The sentence is よし、風呂に入るか！. In the video, it is near the end. Or if you don't want to search blindly for it, it is the 3rd from the bottom in the script here.

Is the meaning of か in that sentence this one?

1. ７ 引用した句の意味やある事実を確かめ、自分自身に言い聞かせる意を表す。「急がば回れ―」「そろそろ寝るとする―」

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か［副助終助並助係助］の意味 - 国語辞書 - goo辞書
In other words, the sentence (よし、風呂に入るか！) isn't directed at Erin but at the speaker himself (the father), right?

Yes, that's right.

Thank you!