Here’s my translation

Alan

Gríðr svarar: "Engan eld fær þú af mér, nema þú mælir þrjú sannyrði, ok ef þú gerir þetta skjótt, þá skaltu liggja hjá dóttur minni, en ef þú vilt eigi þenna kost, þá hirði ek eigi, þó at Björn kali til dauðs." 
Gríðr answers: “You [will] get no fire from me, unless you speak three true-words, and if you do this quickly, then you-shall lie with my daughter, but if you want not this choice, then I care not, even though Björn should-freeze to death.

Illugi segist þetta vilja.
Illugi says-of-himself to want this.

Eptir þetta gengr fram ein kona. 
After this a certain woman steps forward.

Hún er svá fríð, at Illugi þóttist enga fríðari sét hafa. 
She is so beautiful, that Illugi bethought-himself to have seen no-one more-beautiful.

Ok þegar er hann sá hana, felldi hann mikla ást til hennar. 
And as-soon-as he saw her, he took a great-fancy to her.

Hún var hljóð ok fáræðin. 
She was taciturn and chary-of-words (of few words).

Illugi tók þá til máls: "Mér mun mál," segir hann, "at fá sannyrðin: Hellir þínn er hár ok breiðr, at ek hefi eigi sét hús meira né sterkara. 
Illugi took then to speech: “[It] will [be] high-time (<mál>, Z.ii.2) for me,” says he, “to deliver (<fá>, Z6) the-true-words: Your cave is tall and broad, [such] that I have not seen a house larger nor stronger.

Svá er ok nefit á þér mikit, at ek hefi eigi sét meira skrípi en þú ert, ok svá svört, at fagrt er gólfit hjá þér, ok enga hefi ek ámátligri sét en þik, ok víst er dóttir þín fegri, ok ykkar hefi ek sét mestan mun, ok svá munu allir segja, er ykkr sjá." 
Also the nose on you (ie your nose) is so prominent, that I have not seen a greater grotesque-monster than you are, and so black that the-floor is fair in-comparison to you, and I have seen no-one more-loathsome than you, and certainly your daughter is fairer, and I have seen [the] greatest difference between you-two (ie mother and daughter – chalk and cheese). And so will all say, who see you-two.”

Gríðr mælti þá: "Víst er þat," segir hún, "at hvárki viltu lofa mik né hæla mér, ok ekki lízt þér svá illa á mik sem þú lætr, en þat þykki mér nú ráð, at þú farir í hvílu með dóttur minni, ok leik allt þat, er þik lystir, því at þér lízt betr á dóttur mína en á mik. 
Gríðr spoke then: “That is certain,” says she, “that you-want neither to laud me nor praise me, and [it] seems not to you so badly about me as you express (ie, you don´t think so badly of me as you let on, <láta>, Z10) , but (and) that seems to me now [the] advisable-thing, that you go to bed with my daughter, and perform-[you, imperative] all that, which pleases you, because [it] seems to you better regarding my daughter than regarding me (ie you seem to prefer my daughter to me).

Nú mun skjótt at öllu farit," segir Gríðr, "ok þarf ekki lengi lýsingar." 
Now [it] will [be] passed speedily in all respects (<at öllu>, allr, Z5) ,’ (ie events will come to a head quickly) says Gríðr, “and [it] wants not long till daybreak.” (ie daybreak is not far off – with the implication: so do not dally -, but maybe with the double-meaning: ‘the marriage-banns have not long to wait.’)

Illugi segir svá vera skyldu. 
Illugi says so [it] should be.

Gengr hann at hvílunni ok kastar klæðum, en kerling þjónar dóttur sinni, ok kómu þau í eina hvílu bæði. 
He goes now to the-bed and casts-off [his] clothes, but (and) [the] old-woman (ie Gríðr) waits-on (attends on) her daughter, and they both (ie Illugi and the daughter) come into [the] one bed.

Illugi snýst at henni ok gerir sik blíðan, en hún gerir enga gleði af sér. 
Illugi turns-himself (<snúast>, Z8) towards her and becomes (lit: makes himself) agreeable, but she gives no joy from herself (ie she doesn´t reciprocate).

Þrífr Gríðr þá í hár Illuga ok kippir honun fram á stokkinn, en annarri hendi brá hún björtu saxi ok mjök bitrligu ok reiddi at höfði honum, en Illugi lá kyrr ok hrærði hvergi á sér.
Gríðr then lays-hold of [the] hair of Illugi and pulls him forward onto the-board-at-the-front-of-the-bed, but (and) with [the]-other hand she draws a shining short-single-edged-sword and very sharp and brandished [it] at [the] head of him (typo, ie Illugi), but Illugi lay still and bestirred himself not-at-all.