The midi works for me; perhaps the problem is at your end. So far as a tune for the broadside copy goes, this is one case where the traditional variants are still recognisably close to the 18th century tune (see below).

Some information to add:

Note: verse 4 in As Sylvie Was Walking in the DT contains an error of transcription. Line 3 should be:

He's a-gone and he's leaved me, he's a-gone, he's deceived me

As stated in the Penguin Book notes, the text given is a collation from three separate sources; in fact only verse 1 is given as noted from Mrs. Aston (who was the collector's mother, incidentally). Her text was as follows:

SYLVIA

(Noted by Tilly Aston, June 27th, 1911, from Mrs. Aston, Moonee Ponds, Melbourne, Australia)

As Sylvie was walking down by the riverside,
As Sylvie was walking down by the riverside,
And looking so sadly, and looking so sadly
And looking so sadly upon its swift tide.

She thought on the lover that left her in pride;
On the banks of the meadow she sat down and cried,
On the banks of the meadow, on the banks of the meadow
On the banks of the meadow she sat down and cried.

And there she sat crying, when her young man came by,
And he said, "My dear jewel, what makes you to cry?"
And she said, "I am vexed, love, and troubled in mind,
Through an unconstant lover who proved so unkind."

"Come, dig me a grave that is long, wide, and deep,
That I may lie down there and take a long sleep,
And strew it with laurels, and strew it with laurels,
And strew it with laurels and posies so sweet."

The above from The Journal of the English Folk Dance and Song Society, vol.I no.1, 1932. Mrs. Hann and Mrs. Russell's versions appeared in The Journal of the Folk Song Society, vol.3, issue 11, 1907. The additional material incorporated into the Penguin text breaks down as follows:

Verse 1:  Mrs. Aston
Verse 2:  Mrs. Aston (edited)
Verse 3, line 1:  Mrs. Aston
Verse 3, line 2:  Mrs. Aston (edited)
Verse 3, lines 3 and 4:  Mrs. Hann (from her first verse)
Verse 4:  Mrs. Hann (from her third verse, slightly edited)
Verse 5:  Mrs. Russell (her first verse)
Verse 6:  Mrs. Russell (her second verse)

Verse 7 is quite another matter, and comes, not from any of the three ladies mentioned, but from a set collected by Sabine Baring Gould from James Parsons of Lew Down, Devonshire, in October 1888; the editors have rewritten it to fit the pattern of Sylvie. Mr. Parsons sang it thus:

I'll spread my sail of silver,
I'll steer toward the sun,
And thou, false love, will weep for me,
And thou, false love, will weep for me,
For me when I'm gone.

Mr. Parsons' text as originally noted appears in James Reeves, The Everlasting Circle (Heinemann, 1960):

THE FORSAKEN MAIDEN

(Noted by Sabine Baring Gould from James Parsons, Lew Down, Devon, 1888)

A maiden sat a-weeping
Down by the sea-shore.
What ails my pretty Sally,
What ails my pretty Sally
And makes her heart sore?

Because I am a-weary
A-weary in my mind.
No comfort and no pleasure,
No comfort and no pleasure
Henceforth can I find.

I'll spread my sail of silver,
I'll loose my rope of silk.
My mast is of the cypress tree,
My mast is of the cypress tree
My track is white as milk.

I'll spread my sail of silver,
I'll steer toward the sun,
And thou, false love, will weep for me,
And thou, false love, will weep for me,
For me when I'm gone.

The song appeared, as A Maiden Sat A-Weeping, in Baring Gould's Songs of the West, with some unnecessary editorial changes which need not concern us here, beyond the fact that in each verse the third line is repeated three times rather than two, as above. Here is a midi of Mr. Parsons' tune, made from the notation in that book:

The Forsaken Maiden (midi)

Baring Gould thought the tune to be 16th century, but later opinion would place it in the early 18th. Stephen Sedley (The Seeds of Love, 1967) states: "The text of this song... first appears in [the ballad-opera] The Beggar's Wedding (1729), though it is not known if it originated there". (According to Bruce Olson, 1729 was actually the date of the 4th edition.) Sedley's text is a collation made from Mrs. Hann's set and a broadside example; he does however also quote an 18th century example of the tune, from Wright's Complete Tutor For Ye [sic] Flute, c.1733. Here is a midi:

Once I Had a Truelove: c.1733 (midi)

The song is number 170 in the Roud Folk Song Index.

Other versions were noted by Cecil Sharp, and it was still turning up in tradition in the 1960s and '70s; both Phoebe Smith and Paddy Tunney had sets of it, and another, from Edith Smith, a servant at Coleby Hall in Lincolnshire (via Maurice Ogg) was printed in English Dance and Song vol.43 no.2, 1981.