Here´s my translation. Glad to hear you are safe, Grace. Sorry for the delay. I´m still having computer problems but fingers crossed the worst is behind me.

Kveðja

Alan

Ásgeir æðikollur átti fimm börn. Son hans hét Auðun faðir Ásgeirs, föður
Ásgeirr Hot-head had five children. His son was-called Auðunn, father of Ásgeirr, father

Auðunar, föður Egils er átti Úlfeiði dóttur Eyjólfs hins halta. Þeirra son
of Auðunn, father of Egill who had (in marriage) Úlfeiða, daughter of Eyjólfr the Halting (Lame). Their son

var Eyjólfur er veginn var á alþingi. Annar son Ásgeirs hét Þorvaldur. Hans
was Eyjólfr who was slain at (the) Althing. (The) second son of Ásgeirr was-called Þorvaldr. His

dóttir var Dalla er átti Ísleifur biskup. Þeirra son var Gissur biskup. Hinn
daughter was Dalla whom Bishop Ísleifr had (in marriage). Their son was Bishop Gissur. The

þriðji son Ásgeirs hét Kálfur. Allir voru synir Ásgeirs vænlegir menn.
third son of Ásgeirr was-called Kálfr. All Ásgeirr’s sons were fine (promising) people (men).

Kálfur Ásgeirsson var þann tíma í förum og þótti hinn nýsti maður.
Kálfr Ásgeirr’s-son was at that time on trading-voyages (see för, Z1) and was-reckoned the freshest (newest?, superl of nýr?) person.

Dóttir Ásgeirs hét Þuríður. Hún var gift Þorkeli kugga syni Þórðar gellis.
(The) daughter of Ásgeirr was-called Þuriðr. She was given-in-marriage (gifted) to Þorkell ‘Cog,’ son of Þorðr Roarer.

Þeirra son var Þorsteinn. Önnur dóttir Ásgeirs hét Hrefna. Hún var vænst
Their son was Þorsteinn. (The) second daughter of Ásgeirr was-called Hrefna. She was (the) most-beautiful

kvenna norður þar í sveitum og vel vinsæl. Ásgeir var mikill maður fyrir sér.
of women, north there in (the) districts (plural) (in the northern districts) and really popular. Ásgeirr was a powerful man of himself (in his own right, see fyrir, Z.i.13).


Það er sagt eitt sinn, að Kjartan Ólafsson byrjaði ferð sína suður til
That is said, one time, that Kjartan Ólafr’s-son set-out-on his journey south to

Borgarfjarðar. Ekki er getið um ferð hans fyrr en hann kom til Borgar. Þar
Borgarfjörðr.(gen. sg)  Nothing is recorded about his journey before he came to Borg. There

bjó þá Þorsteinn Egilsson móðurbróðir hans. Bolli var í ferð með honum því
lived then (at that time) Þorsteinn Egill´s-son, his mother’s-brother. Bolli was on the journey with him because

að svo var ástúðigt með þeim fóstbræðrum að hvorgi þóttist nýta mega að þeir
(it) was so amicable (literally) between those  sworn-brothers (ie they were on such friendly terms) that neither bethought-himself able to bear (it) that they

væru eigi ásamt. Þorsteinn tók við Kjartani með allri blíðu, kvaðst þökk kunna
were not together. Þorsteinn received Kjartan with all friendliness, declared-of-himself to know (feel) pleasure

að hann væri þar lengur en skemur. Kjartan dvelst að Borg um hríð.
that he would-be (stay) there longer (rather) than shorter. Kjartan stayed at Borg for a while (which means so did Bolli).

Þetta sumar stóð skip uppi í Gufuárósi. Það skip átti Kálfur Ásgeirsson.
This summer a ship was-laid up (ashore) in Gufuárós (Steam-River-Mouth). Kálfr Ásgeirr’s-son had (owned) that ship.

Hann hafði verið um veturinn á vist með Þorsteini Egilssyni. Kjartan segir
He had been during the-winter on a stay (stop-over) with Þorsteinn Egill’s-son. Kjartan says

Þorsteini í hljóði að það var mest erindi hans suður þangað að hann vildi
to Þorsteinn on-the-quiet that that was his main mission (objective) south thither that he wanted

kaupa skip hálft að Kálfi, "er mér á því hugur að fara utan" og spyr
to buy a ship half (a half-share in the ship) from Kálfr (see at, Z.iv.8), “(the) wish is for me in that to travel abroad” and asks

Þorstein hversu honum virðist Kálfur.
Þorsteinn how Kálfr (nominative) seemed to him (dative).

Þorsteinn kvaðst hyggja að hann væri góður drengur "er það vorkunn (várkunn) mikil
Þorsteinn declared-of-himself to believe that he was a good (noble-minded) man “That is a great thing-to-be-excused,

frændi," segir Þorsteinn, "að þig fýsi að kanna annarra manna siðu.
kinsman,” says Þorsteinn, “that (it) urges you (ie that you are eager) to search (explore, kanna not kunna) the customs (ways) of other men.

Mun þín ferð verða merkileg með nokkuru móti. Eiga frændur þínir mikið í hættu
Your journey will be remarkable in some way (degree, see mót, Z4). Your kinsmen have much at risk (ie for them much depends on)

hversu þér tekst ferðin."
how the-journey succeeds (turns out) for you.”

Kjartan kvað vel takast munu.
Kjartan declared (they?) will succeed well.