Why does Hrothgar ask Beowulf to battle Grendel's mother, and what does Beowulf do after he kills her?
Grendel's mother is grieving for the loss of her son, so she seeks revenge and attacks Herot. Hrothgar's closest friend is carried off by her into her deep underground/water lair. Along with his friend, Grendel's arm that was hung in Herot Hall was taken by her back to her dwelling.
Hrothgar goes to Beowulf and asks for help again in exchange for riches.
"Our only help, again, lies with you...Save us, once more, and again twisted gold, heaped-up ancient treasure, will reward you for the battle you win!"
Once he battles her, he kills her by chopping off her head. He then follows the hall to where the dead Grendel lies and chops off his head, too. He then takes Grendel's head back to Hrothgar and receives his riches.
The death caused by Grendel’s mother was far too close and personal for Hrothgar. Grendel’s mother took Aeschere, a warrior and close confidant of the king. Grendel’s mother was on a mission to avenge her son’s death in Heorot, and she escaped with her son’s arm, which was on display at the mead house. Hrothgar asked Beowulf to battle Grendel’s mother because the king believed he was the only man capable of dealing with such monsters. This was because Beowulf had earlier succeeded in mortally wounding Grendel. Further, Hrothgar wanted revenge for his close friend.
BEOWULF, SON OF Ecgtheow, spoke: “Do not lament, wise sire! It seems better that each man avenge his friends than to mourn them to no end."
After killing Grendel’s mother using a supernatural sword forged by giants, Beowulf sought Grendel’s body and severed its head. Beowulf made it out of the water carrying Grendel’s head and the sword’s hilt, which was the only part left after the blade supernaturally wasted away. Beowulf and his men presented Grendel’s head on a spear to Hrothgar.
Then that prince of thanes [Beowulf] proceeded to enter; the fearless fighting man of wide renown, the valiant hero, went to greet Hrothgar. After him, Grendel's head was borne into the hall where the men were drinking; it struck awe in the clansmen and queen alike, and the men looked upon this monstrous visage...