Vlad DeepL has separate pricing for their translation product and for their developer API.
Their translation product can be used for free with limited text lengths and functionality or as a paid version with multiple tiers:
https://www.deepl.com/en/pro
It includes the use of its well made apps.
The prices are not straightforward to compare with Kagi as Kagi offers much more than translation.
On top of that, they have an API that allows DeepL translate to be integrated straight into apps. For example a social media client that will translate posts using DeepL.
This API is free for up to 500000 characters per month or €5.-/month plus usage based character cost:
https://support.deepl.com/hc/en-us/articles/360021183620-DeepL-API-Free-vs-DeepL-API-Pro
Since it is difficult to compare products with different feature sets, here is how I look at it for my own personal needs.
As long as I can use Kagi Translate only via the web interface, its added value is the translation quality which is superior to DeepL.
However, most of the time, DeepL is good enough, so I use it because of the added convenience of a very good app which comes with extensions that integrate with my phone's OS and a desktop App. Its API allows me to use it straight from inside apps.
It takes me much less time to get my translation (albeit with an inferior quality) through DeepL than through Kagi. So I use Kagi translate only when I need the absolute best translation available to me.
Which means for reading things in a language I don't or do not fully understand I use mostly DeepL as it is absolutely good enough to help me understand.
If I am writing and need to make absolutely sure that my final text will be understood the best possible way in a foreign language I switch to Kagi.
I understand this takes time to build and has a cost attached to it and I would not expect this to be part of my Kagi subscription.
I would happily pay an extra €10/month for Kagi translate with OS integration through apps and extensions.
Currently I pay about a total of €8/month for the way I use DeepL.