Alannah Kate - 9 out of 10. This is a very pretty name. It looks pretty, too. I've never heard of the name Alannah, but I really like it. And because it has three syllables, I think it goes perfectly with Kate; being a one-syllable name.
Amelie Kate - 10 out of 10! Amelie has been one of my favorite girl names for a while now. It's French, and is usually spelled Amélie. Again, I think it flows beautifully with Kate being the middle name because of it being a three-syllable name.
Eilidh Kate - 8 out of 10. I wasn't sure of how to pronounce this at first, but then I saw you gave the pronunciation. Haha. It's very pretty, but if you don't live in Scotland, I'm afraid that many people will mispronounce her name. But if that isn't a problem for you, then Eilidh Kate is a beautiful name.
Helena Kate - 10 out of 10! I adore the French pronunciation of this name! It's gorgeous. So elegant and sophisticated. I love, love, love it. And I think it pairs beautifully with Kate as the middle name. This is a really lovely name, actually my favorite on this list!
Lois Kate - 6 out of 10. I'm not sure about this name. I'm leaning more towards the 'Don't Like' side. It reminds me of Family Guy, haha. Because the mother is named Lois. I've just never liked the name, though. I don't know what about it I don't like. It sounds kind of old lady-ish, if that makes sense.
Miriam Kate - 7 out of 10. This is an alright name, definitely not as beautiful as the others you have listed. It isn't my least favorite off of the list, though. Miriam was the name of my aunt's dog, so that's the main reason why I wouldn't ever consider naming my child this.
Very nice names, my favorite is Helena Kate! :)