Here’s my translation

Alan

Konungr sagði, at þat skyldi á mánaðarfresti búit vera, --"ok Björn skal fara með þér," segir konungr, "en Illugi vil ek heima sé." 
[The] king said, that should be ready with a month´s-notice (ie in a month), - „and Björn shall journey with you,“ says [the] king, „but I want [that] Illugi be (stay) at-home.

Sigurðr svarar: "At vísu vil ek, at Illugi fari." 
Sigurðr answers: “Certainly I want, that Illugi journey.”

En konungr sagði Björn skyldi fylgja honum, --"því at hann er hverjum kappa meiri ok bilar aldri í stríði. 
But [the] king said [that] Björn (accusative) should accompany him, - “because he is greater than (superior to) every man-of-valour (dative with comparative) and never gives way in adversity.

Hann mun þér vera hollr ok trúr, sem hann hefir mér verit," sagði konungr, ok skilja þeir nú tal sitt. 
He will be loyal and faithful to you, as he has been to me,’ said the king, and they break-off now their conversation.

Eptir þetta gengr konungsson til Sviða ok segir Hildi tal þeira feðga. 
After this, [the] king’s-son goes to Sviði and says to Hildr [the] conversation of them, father-and-son.

Hún segir son sinn ungan vera ok eigi í hernað fara mega. 
She says her son (accusative) to be young and not able to go a-harrying.

"Er hann ok ekki reyndr," segir hún, "vilda ek ok ekki heldr, at Björn brygði honum því, at hann þyrði ekki at berjast með þér í orrostu." 
“He is also not experienced (proven),” says she, “I would also rather not, that Björn upbraided him for that, that he dared not to fight with you in battle.”

Lýkr Hildr svá máli, en konungsson ferr heim til hallar ok er mjök óglaðr.
Hildr concludes thus [the] speech (interview), but (and) the king’s-son journeys home to [the] hall and is very gloomy.

Ambátt sú var hjá Hildi, er Sunnlöð hét. 
That bondwoman was with Hildr, who was-called Sunnlöð.

Hún var fjölkunnug ok in mesta kvöldriða. 
She was skilled-in-magic and the greatest night-hag.

Hún hafði margan mann illa leikit. 
She had ill-treated (presumably by bewitching them) many a person.

Hildr kemr at máli við Illuga ok biðr hann sækja pál í sel, er Sviði lét eptir. 
Hildr comes to a talking (ie to talk) with Illugi and bids him to go-fetch a hoe (or spade) in [the] shieling (mountain-pasture-shed), which Sviði left behind.

Hann játar því. 
He agrees (says yes) to that.

Þat var síð dags, er Illugi gekk heiman. 
That was late in [the] day, when Illugi went from-home.

Hann fór hart ok kom til seljanna ok fann þar pálinn. 
He journeyed fast (cf. <harðr>, Z3) and came to the-shieling and found there the-hoe (spade).

Þá var myrkt af nótt, ok ferr þó frá selinu, ok er hann var skammt kominn, var hlaupit á bak honum svá hart, at hælarnir kómu framan á bringuna. 
Then (ie by that time) [it] was dark of night (ie the night was dark), and [he] journeys from the-shieling, and when he was (had) come a short-distance, the-leaping-assault on his back was so hard, that the-heels (of the attacker) came forward on to the-chest (of the attackee).

Þetta kvikendi hafði vönd í hendi ok barði Illuga með. 
This living-creature had a ‘wand’ (or switch) in hand and beat (<berja>, Z1) Illugi with [it]

Hér var komin Sunnlöð.
Here was (had) come Sunnlöð.

Illugi gengr ekki at síðr ok bar flagð þetta langa leið, þar til hann kemr at einum stórum steini. 
Illugi walks [on] not less (none-the-less) and carried this (<Þetta> is neut acc sg agreeing with <flagð> ) ogress a long way, until he comes to a certain large stone.

Hann keyrir flagðit niðr við steininn svá hart, at hryggrinn brotnar, ok lét hún svá líf sitt. 
He flings the-ogress down against the-stone so hard, that the-backbone is-broken, and she gave-up thus her life.

Létti hann eigi sinni ferð fyrr en hann kemr heim. 
He stopped not his journey before he comes home.

Hildr, móðir hans, var úti, er hann kom heim. 
Hildr, his mother, was outside, when he came home.

Illugi var þá ófrýnn. 
Illugi was then frowning (unpleasant-looking).

Hildr var þá blíð. 
Hildr was then pleasant (affable)

"Hafa nokkur nýtíðendi gerzt í þinni ferð, sonr minn?" sagði hún. 
“Have some new-events (note: gramm plural) arisen in your journey, my son?” said she.

"Fannstu pálinn, sem ek vísaði þér á?" 
“Found-you the-hoe, which I indicated to you?”

"Já," segir Illugi. 
“Yes,” says Illugi.