As you know, most of the literature on "medieval" Iceland tend to focus on the classical "Free State" Period before 1262/63.

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(Primary Sources)

• Einarr Haflidason. The Life of Laurence, Bishop of Hólar in Iceland (Laurentius saga), trans. Oliver Elton. London, 1890: The best (near) contemporary source for the 1st half of the 14th century in Iceland in fact has an old English translation. If your professor insists on the original edition, the following is the standard one in normalized Old Norse: Guðrún Ása Grímsdóttir (útg.), Biskupa sögur III: Árna saga biskups, Lárentius saga biskups, Söguþáttr Jóns Halldórssonar biskups, Biskupa ættir. Reykjavík: Hið Íslenzka fornritfélag, 1998. ÍF XVII. You might also have to check the deeds of Bishop Árni Þorláksson of Skálholt in Árna saga biskups that has been translated in modern Norwegian.
• Another important source of 14th century Iceland is documentary sources (charters, especially of church donation, called máldagi) published in Diplomatarium Islandicum series. Earlier volume of this series of documentary sources is to be found in the scanned ones (copyright expired) in Internet Archive.
• Law sources for the Later Middle Iceland:
• Schulman, Jana K. (ed. & trans.). The Laws of Later Iceland: Jónsbók. Saarbrücken: AQ, 2010.
• Synodal statutes of the Norwegian church province of Nidaros/Trondheim (while not specially codified for the Icelandic church, the extant text is taken mainly from the 14th Icelandic manuscript) is found in Norges gamle love, bd. 3: Lovgivningen efter Kong Magnus Haakonssøns Død 1280 indtil 1387, utg. Rudolf Keyser & Peter A. Munch, Christiania, 1849, ss. 227-310. Only one of these statutes has been translated (in modern Norwegian).
• Storm, Gustav (útg.). Islandske Annaler indtil 1578. Christiania, 1888: includes annals written in the 13th and 14th century Iceland (copyright expired so that we can have access online). https://baekur.is/bok/000376834

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(Useful for OP's purpose?)

• Agnes S. Arnórsdóttir. Property and Virginity: The Christianization of Marriage in Medieval Iceland 1200-1600. Aarhus: Aarhus UP, 2010: analyzes the marriage contract documents in later medieval Iceland in long-term trend.
• Falk, Oren. Violence & Risk in Medieval Iceland: This Spattered Isle. Oxford: OUP, 2021: This new book features risk managements against several kind of dangers, and it is very unique also with focus on the volcanic activity in medieval Iceland in its chapter 5 (pp. 221-76).
• Gelsinger, Bruce E. Icelandic Enterprise: Commerce and Economy in the Middle Ages. Columbia, SC: U of South Carolina Pr., 1981: is the classic of medieval Icelandic economic history, but mainly only "the epilogue" (pp. 181-94) deals with post-Free State period. This topic has been explored widely among recent scholarships, so you can instead check some other articles/ book chapters.
• Gunnar Karlsson. The History of Iceland. Minneapolis: U of Minnesota Pr., 2000; Iceland’s 1100 Years: History of a Marginal Society. New Paperback Edition. London: Hurst, 2020: These two are essentially the same, the standard overview book on the History of Iceland in English. While a brief history edition by the same author (linked to goodreads.com) is also certainly available, I'd strongly recommend the longer, standard edition. Unless you can read the series of Icelandic history in modern Icelandic, Saga Íslands, the book is must read to get the basic outline of Icelandic history.
• ________. Lífsbjörg Íslendinga frá 10. öld til 16. aldar. Reykjavík: Háskoláútgáfan, 2009: If you can read modern Icelandic fluently, this must be the book you have to check after the overview book above ([Gunnar Karlsson 2000; 2020]).
• Imsen, Steinar (ed.). The Norwegian Domination and the Norse World c.1100 - c.1400. Trondheim: Rostra, 2010: is one of the series of conferences themed around the Norwegian realm (Norgesveldet) that also incorporated Iceland under the auspice of the king of Norway in High Middle Ages. Boulhosa's chapter deals with the relationship between Icelanders and Norway, mainly from an economic (fishery) point of view.
• ________ (ed.). Rex Insularum: The Kings of Norway and His 'Skattland' as a Political System c. 1260 to c. 1450. Bergen: Fagbokforlaget, 2014: is, so to speak, a concluding volume of Norwegian Dominion/ skattland project (see [Imsen 2010]). 4 chapters in its Part II (pp. 181-321, authored respectively by Jón Viðar Sigurðsson, Lena Rohrbach, Helgi Þorláksson & Randi B. Wærdahl) focus on the relationship between Iceland and Norway mainly from political, administrative, economic and personal network standpoints.
• (open access journal: pdf) Mehler, Natasha, Mark Gardiner & Endre Elvstad (eds.). German Trade in the North Atlantic c. 1400–1700. Interdisciplinary Perspectives. Stavanger: U of Stavanger, 2019. AmS Skrifter 27. NB: mainly focusing on the 15th century and later, based on archaeological evidence.
• Orri Vésteinsson. "Commercial Fishing and the Political Economy of Medieval Iceland." In: Cod & Herring: The Archaeology & History of Medieval Sea Fishing, ed. James H. Barret & David C. Orton, pp. 71-79. Oxford: Oxbow, 2016: This collection of interdisciplinary essays also includes the chapter by Mark Gardinar on the fishery in the 15th century Iceland (sometimes called as "the English Age (enska öldin - linked to the introduction entry found in the site of University of Iceland in modern Icelandic)")

(Any topic on the 14th century Iceland)

• Erika Sigurdson. The Church in Fourteenth-Century Iceland: The Formation of an Elite Clerical Identity. Leiden: Brill, 2016: While it apparently does not deal with OP's concern, this monograph (based on the author's dissertation) makes the best of available (extant) written evidence in the 14th century Iceland.
• Rowe, Elisabeth A. The Development of Flateyjarbók: Iceland and the Norwegian Dynastic Crisis of 1389. Odense: U of Southern Denmark, 2005: combines the philology and historical context, illustrating how changing political trend in the late 14th century could affect the selection of works (thus the entire composition) of one of the most famous medieval Icelandic manuscript, Flateyjarbók.

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(Related Academic Papers)

• Haki Antonsson. “Árna Saga Biskups as Literature and History.” The Journal of English and Germanic Philology 116, no. 3 (2017): 261–85. https://doi.org/10.5406/jenglgermphil.116.3.0261.
• Harrison, Ramona, Howell M. Roberts & W. Paul Adderley. “Gásir in Eyjafjörður: International Exchange and Local Economy in Medieval Iceland.” Journal of the North Atlantic 1 (2008): 99–119. https://www.jstor.org/stable/26663860.
• Jón Viðar Sigurosson. "The Icelandic Aristocracy after the Fall of the Free State." Scandinavian Journal of History 20:3 (1995): 153-166. DOI: 10.1080/03468759508579302
• Ogilvie, Astrid. "Climatic Changes in Iceland A.D. c.865 to 1598." Acta Archaeologica 61 (1991): 233-251. ISSN 0065-101X
• Matsumoto, Sayaka. "A Link with the External World: The Stockfish Trade in 14th-15th century Iceland." Paper given at the First Finnish-Japanese workshop for doctoral students of history and economics (2010).
• Mehler, Natasha. "The Sulphur Trade of Iceland from the Viking Age to the End of the Hanseatic Period". In: I. Baug, J. Larsen, S. Samset Mygland (eds.), Nordic Middle Ages – Artefacts, Landscapes and Society. Essays in Honor of Ingvild Øye on her 70th Birthday. University of Bergen Archaeological Series 8 (Bergen 2015), pp. 193-212. https://bora.uib.no/bora-xmlui/handle/1956/15435
• Steinunn Kristjánsdóttir. "Skriðuklaustur monastery: Medical Centre of Medieval East Iceland?" Acta Archaeologica 79 (2008): 208-15: NB: While this Augustinian cloister in Iceland would found only in the late 15th century, its research on medical practice mainly based on osteological analysis might be useful to get glimpse of everyday life (and death) in medieval Iceland in general.
• Steinunn Kristjánsdóttir, Inger Larsson & Per Arvid Åsen. "The Icelandic Medieval Monastic Garden - Did It Exist?" Scandinavian Journal of History 39-5 (2014): 560-579. DOI: 10.1080/03468755.2014.946534

(Edited): fixes some very silly grammatical mistakes.