Here's some gabc code to get you started. Sounds nice; just a few observations:

(i) I've substituted some note shapes to make it look prettier, especially in the descending scale figures. YMMV.
(ii) "le-i" in "eleison" is usually taken as two syllables.
(iii) Comparing the sung version with the manuscript above, there appears to be a missing note in the manuscript for the last "eleison".
(iv) You might consider a variation for the last repetition of "kyrie eleison".

Agreed. But in the accompanying sound file, it is sung as two. (Unless I am going deaf, which is a possibility). The point that I didn't make very well, is that if you want it sung as two syllables it is helpful to notate it as two.

After reading the handwritten score and transcribing it to GABC, I listened to the recording. Here is the code for that transcription:

name: Kyrie Iesus Nazarenus;
commentary: MMXV;
annotation: i;
%%
(c4) KY(dh)ri(g)e(h.) *() e(ighv.GFEDgde)lei(f)son.(e.) <i>iij</i>() (::)
Chri(h)ste(h.) (,) e(igh.h!ijkvJIHGi)lei(g)son.(h.) <i>iij</i>() (::)
Ky(f)ri(f)e(f.) (,) e(fvEDCegFEDCe)lei(c)son.(d.) <i>iij</i>() (::)

While the recording did not quite seem to match the written score (as transcribed by me), I felt that there indeed needed to be a separate syllables for the "lei" in "eleison"; i.e. "e-le-i-son" rather than "e-lei-son". However, my take on modifying the score is different from Gregory's. So I came up with the following solution: move "le-" to the note preceding the former "lei-" syllable (which now becomes the "i-" syllable):

name: Kyrie Iesus Nazarenus;
commentary: MMXV;
annotation: i;
%%
(c4) KY(dh)ri(g)e(h.) *() e(ighv.GFEDgd)le(e)i(f)son.(e.) <i>iij</i>() (::)
Chri(h)ste(h.) (,) e(igh.hiwjkvJIHG)le(i)i(g)son.(h.) <i>iij</i>() (::)
Ky(f)ri(f)e(f.) (,) e(fvEDCegFEDC)le(e)i(c)son.(d.) <i>iij</i>() (::)

By this point, I felt that the "Christe" and second "Kyrie" were not quite right (to my own taste), too short; hence, I modified the code even more to produce the following:

name: Kyrie Iesus Nazarenus;
commentary: MMXV;
annotation: i;
%%
(c4) KY(dh)ri(g)e(h.) *() e(ighv.GFEDgd)le(e)i(f)son.(e.) <i>iij</i>() (::)
Chri(hih)ste(h.) (,) e(igh.hiwjkvJIHG)le(i)i(g)son.(h.) <i>iij</i>() (::)
Ky(h)ri(ef)e(f.) (,) e(fvEDCegFEDC)le(e)i(c)son.(d.) <i>iij</i>() (::)

Finally, after singing through the results several times, I felt that the "le-" syllable (which has the accent or emphasis in pronunciation of "eleison") should carry the bulk of the melisma that the "e-" syllable had been given. So I moved "le-" up in each case and added one more note to the first syllable of the second "Kyrie", obtaining the following (which I like best of all):

name: Kyrie Iesus Nazarenus;
commentary: MMXV;
annotation: i;
%%
(c4) KY(dh)ri(g)e(h.) *() e(igh.)le(hvGFEDgd)i(ef)son.(e.) <i>iij</i>() (::)
Chri(hih)ste(h.) (,) e(igh.)le(hiwjkvJIHG)i(ig)son.(h.) <i>iij</i>() (::)
Ky(hg)ri(ef)e(f.) (,) e(fvEDC)le(egFEDC)i(ec)son.(d.) <i>iij</i>() (::)

The four attached PDFs give, in order, each of these versions. They represent my own thoughts on possible modifications of the original and may be freely used or even modified by those who have still different ideas.