[William Cloud Hicklin]

And hearing the words of Melian Hurin stood moveless, and he gazed long into the eyes of the Queen; and there in Menegroth, defended still by the Girdle of Melian from the darkness of the Enemy, he read the truth of all that was done, and tasted at last the fullness of woe that was measured for him by Morgoth Bauglir.

But Tolkien didn't write that passage. There's no suggestion of any healing at all- this whole business was written by Christopher (in consultation with Guy Kay).

Here however is a passage whose canonicity is if not perfect then still very great:

Who now knows the counsels of Morgoth? Who can measure the reach of his thought, who had been Melkor, mighty among the Ainur of the Great Song, and sat now, the dark lord upon a dark throne in the North, weighing in his malice all the tidings that came to him, whether by spy or by traitor, seeing in the eyes of his mind and understanding far more of the deeds and purposes of his enemies than even the wisest among them feared.

The chessmaster indeed! And it continues

...save Melian the Queen. To her often his thought reached out, and was foiled

In fact, throughout CoH there are mentions of Morgoth's inability to 'see' into Doriath- echoed in Sauron and Galadriel's Lorien. Moreover,

Gondolin still stood, but it was hidden. Doriath he knew, but could not enter yet