Here’s my translation

Alan

Signý mátti ekki mæla fyrir harmi ok gráti. 
Signý could not speak for grief and weeping.

Hildr mælti þá: "Vilda ek, Grímhildr, at ek launaði þér þín álög, ok þat mæli ek um, at öðrum fæti standir þú á skemmu þessari, en öðrum heima á konungshöll. 
Hildr spoke then: “I would-want, Grímhildr, that I rewarded you for your curses (the tribulations you have caused), and I solemnly say (lit: speak about that), that you [will] stand with one foot (dat. sg) in this lady’s-bower, but (and) with [the] other (dat sg, annarr … annarr, Z1) at-home in [the] king’s-hall.

Þrælar skulu þar kynda bál mitt í milli fóta þér. 
Slaves shall there kindle my fire between your feet (legs).

Þat bál skal standa bæði nætr ok daga, ok öll skaltu neðan af eldi brenna, en ofan frjósa, at aldri fáir þú þína ró. 
That fire shall stand (remain) both nights and days (gram plural), and [the] whole you-shall burn from-below from [the] fire, but freeze from-above, [so] that you never get your rest.

En ef vit mæðgur komumst ór þessum álögum, þá skaltu deyja ok detta ofan í bálit." 
But (And) if we-two (<vit>), mother-and-daughter, make-our-way (get through) out-of these curses (tribulations), then you shall die and fall down into the-fire.

Grímhildr tók þá til máls: "Mjök heimskligt er okkart tal, ok vil ek, at þetta haldist hvárki."
Grímhildr took then to talk: “Our conversation is very foolish, and I want, that this (ie Hildr’s fire-pledge) should-stand (ie be valid) neither [(the one nor the other) part]. (<hvárki> is neut, so can’t refer to the women)] (ie that neither of the two parts of Hildr´s pledge should stand, be valid, come-to-pass).

Hildr sagði, at þat yrði at standa. 
Hildr said, that that (ie her pledge) needs-must stand.

Hurfu þær mæðgur þá burt í helli þenna, ok er ek sú in sama Signý, ok er hér Hildr, dóttir min, ok vil ek nú gifta þér hana ok launa þér svá, at þú hefir mér ór álögum komit." 
They, mother-and-daughter, disappeared away into this cave, and I am now she, the same Signý, and here is Hildr, my daughter, and I want now to give (in marriage) her to you and to reward you for such, that you have brought (<koma> + dat, Z3) me out-of [the] curses (tribulations)

Ok at endaðri þessari sögu koma í hellinn sjau skessur með bitrligum skálmum ok hlaupa at Gríði ok höggva til hennar bæði hart ok tíðum. 
And at this concluded story (fem dat sg) seven giantesses came into the cave with sharp short-swords and leap at Gríðr and hewed at her both hard and often.

Hildr var nú ákafliga hrædd. 
Hildr was now exceedingly afraid.

Illugi veitir Gríði dugnað ok höggr til þeira bæði ótt ok tíðum, ok eigi léttir hann fyrr en þau hafa drepit þær allar, ok brennir þær allar á báli. 
Illugi gives to Gríðr assistance and hews at them both vehemently and often, and he stops not before they have killed them all, and [he] burns them all in [the] fire.

Gríðr mælti þá: "Nú hefir þú, Illugi, frelst okkr bæði af þessum skessum, ok hefi ek við þær átt ellifu vetr." 
Gríðr spoke then: “Now you have, Illugi, delivered us both from these giantesses, and I have fought (<eiga við e-n>, Z10) with them eleven winters.”

Illugi segir þat nógu lengi verit hafa.
Illugi says that to have been long enough.

6. Sigurðr fekk Signýjar
6. Sigurðr won Signý

Eptir þetta fylgir Gríðr þeim til báts Illuga, ok gaf hún þeim gull ok marga dýra gripi, ok hafði hann nú með sér eldinn, ok skildist hún þar við þau. 
After this Gríðr accompanies them to [the] boat of Illugi, and she gave them gold and many precious treasures, and he had now with him the-fire, and she parted-company with them there.

Illugi rær aptr til sinna manna. 
Illugi rows back to his people (men).

Þeir urðu við þat glaðir ok létu sér hitna. 
They became (were) with that glad and caused themselves to become-hot (warmed themselves courtesy of the fire)

Mánuð lá konungsson þar, ok gaf honum aldri byr. 
A month [the] king’s-son lay (was-situated) there, and [it] never gave him fair-wind.

Björn kenndi þat Hildi ok kvað Illuga hana hafa sótta í hella, ok segir Björn, at hún sé in mesta tröllkona. 
Björn imputed (<kenna e-m e-t>, Z4) that (ie the lack of fair wind) to Hildr and declared Illugi to have sought (<soekja>) her in caves (acc pl), and Björn (nom) says that she is [the] greatest troll-woman.

Sigurðr bað Björn þegja, ok ekki vildi hann því trúa, er Björn sagði. 
Sigurðr bade Björn to-be-silent, and he wanted not to believe that, which Björn said.