Torah as the eternal source of joy

ומה שעושין שמחת תורה עתה.

As for why we make Simchas Torah now —

The Sefas Emes opens by asking why the rejoicing of Simchas Torah is set specifically here, at the close of the festival.

לפי פשוטו כי עתה אין לנו שמחת החג וניסוך המים ואין לנו שיור רק התורה.

On the simple level: now we no longer have the simchah of the Chag and the nisuch hamayim (water-libation), and we are left with nothing but the Torah.

Once the avodah of Sukkos in the Bais Hamikdash has ended, the only thing that remains in our hands is the Torah itself, so the rejoicing now attaches to it.

ובאמת בתורה יש לנו כל זאת כמ"ש זאת תורת המנחה חטאת דרשו חז"ל כל העוסק בתורה כו'.

But in truth, in the Torah we still have all of this, as it is written, "This is the Torah of the minchah... of the chatas" — Chazal expounded: "Whoever toils in Torah [it is as if he brought all the korbanos]" (Menachos 110a).

The Torah is not merely what is left over; it actually contains within it everything that was lost. Learning Torah accomplishes what every korban once accomplished, so nothing is truly missing.

ולכן בכח התורה יכולין לעורר כל שמחת החג.

Therefore, through the power of the Torah we are able to reawaken all the simchah of the Chag.

Because the Torah holds within it the inner reality of the whole festival, learning it can rekindle the very joy of Sukkos even after the Chag itself has passed.

וזהו ג"כ סימן שלבנ"י נשאר השמחה לעולם ולאומות הי' רק לשעה.

And this is also a sign that for Bnei Yisrael the simchah remains forever, whereas for the nations it was only for a moment.

Israel's joy is rooted in the eternal Torah, so it never expires; the nations' kind of joy is tied to the passing moment and disappears with it.

וכן אמרו חז"ל כי האומות אבדו בחורבן ביהמ"ק יותר:

And so Chazal said that the nations lost more in the churban of the Bais Hamikdash [than Israel did].

Even the destruction proves the point: the nations forfeited the lasting benefit that the Mikdash brought the world, while Israel still holds the Torah, the source of an enduring joy that no churban can take away.

Summary: After the festival ends, only the Torah remains in our hands — yet the Torah contains the whole reality of the Chag, so through it we reawaken all the simchah of Sukkos. This is why Israel's joy is eternal while the nations' is only fleeting.