"In which scriptures is there mention of Ashwini Kumaras being taught the secrets of Ayurveda - the Ancient Indian Medical Science." ?
There are two schools of medicine one is Atreya School which is represented by Charaka and the second is the Dhanvantari School of surgery represented by Shushruta.
Brahma who is said to be the first teacher of the universe originally composed the Ayurveda. Having propounded the science of healing, brahma propagated this knowledge through Daksha Prajapati, who taught this knowledge to Ashwini Kumaras.
But in south Rishi Agastya is credited with knowledge of Ayurveda.
The answer of your question is given in Charak Samhita of Ayurveda. It's described in Chapter one – Sutra Sthanam – Shloka No. 4-5 that ,Daksha Prajapati gave this knowledge to Ashwini Kumaras. In these shlokas 4-5 , Hierarchy of transformation of knowledge in Ayurveda is described .
ब्रह्मणा हि यथा प्रोक्तमायुर्वेदं प्रजापति: |
निखिलेनाssदावश्विनौ तु पुनस्तत: ||4||
प्रतिपेदेह केवलं |
| ऋषिप्रोक्तो भरद्वाजतस्माच्छक्रमुपागमत्
brahmaṇā hi yathāprōktamāyurvēdaṁ prajāpatiḥ|
nikhilēnādāvaśvinau tu punastataḥ||4||
pratipēdē ha kēvalam|
In the beginning Lord Brahma taught this knowledge which Prajapti
Daksha fully acquired. From Daksha two Ashwini Kumaras acquired this
knowledge. And from Ashwini Kumaras Indra received this knowledge.
So as to get this knowledge Rishi Bharatdwaja came to Indra.
At first, Prajapati received Ayurveda in its entirety as recited by Brahma, from whom the Ashwins (received), and from Ashwins lord Indra received it fully. That is why Bharadwaja , as beseeched by the sages, came to Indra.[4-5]