Is it normal to like this type of music?

As long as its not mainstream pop.

Yes it is normal. You can like Christian songs even if you're another religion or lack a religion.
Music can appeal (e.g. sound, lyrics, melody, etc.), and has different meanings to everyone. A lot of her music is about hope, faith, and strength and perseverance. They have a universal appeal regardless of religion. There's no reason to feel guilty.

On a side note, I like Britt Nicole too. Check out Krystal Meyers, Flyleaf and Fireflight (they're in similar genres and have a lot of variety in songs you might enjoy).

I listen to Christian rock bands and I don't even believe in god. As long as they make good music and are not too preachy I'm fine with it.

Its music. Don't worry, and don't feel guilty. You like it because it probably sounds amazing, it doesn't mean that you've stopped believing in what you always believed in.

You probably really enjoy the styling of the music and hopeful lyrics. You don't have to be Christian to enjoy it. There are a lot of good Christian rock songs. I've liked quite a few, most of them I had no idea that they were taking about god.

We can't choose what music we love. Don't worry. The older I get, the less I care about what people think of my taste in anything. Just embrace it, it's there to be listened to and enjoyed.

I listen to a lot of Christian music too, even though I don't really believe in most of the concepts of mainstream Christianity. My mother banned secular music while I was growing up, so Christian was my only choice. What can say, a lot of the artists from my childhood stuck with me.

I wouldn't worry about it. Even though I'm atheist, gospel music often gives me chills. Gregorian chants and native American music are also examples of music people listen to for relaxation, regardless of their own beliefs. One of my favourite singers often sings about her Christianity which seems to be a great source of inspiration to her.