Here it goes -
March 13th - Stay overnight at Hotel Berg near KeflaviK
March 14th - You have two options : (Keep in mind, public transport in Iceland in non existent, so rely mostly on booking tours or renting a car. Taxis are available but expensive.
1) Go to Blue Lagoon on your way to Reykjavik
Blue Lagoon is in Grindavik near Keflavik (~15 kms) so makes sense to go there either on your way from airport or on your way to airport. I would classify this as a must do place and you can buy tickets online or at Blue Lagoon itself (long lines expected). If you want a massage, def book beforehand. This is a geothermal spa where they pump water up from geothermal springs underneath.
Many tour buses pick you up from airport/hotel near airport and drop you at blue lagoon, pick you up from blue lagoon and drop you at your hotel in Reykjavik (~15 Pounds). You can also hail a taxi but those are more expensive for the whole tour (~75 pounds). reykjavik excursions, iceland excursions, flybus, tour.is (many tour operators do this). World recommend booking this tour online as you do not want to be hassled that day. These buses also store yoru luggage for you while you are at Blue Lagoon.
2) Go to Reykjavic directly
Same tour companies will take you to reykjavic directly. Keep in mind, the buses are an an interval so would recommened planning accordingly. I saw Iceland, Reykjavik excursions stand in the airport terminal if you wanted to book on arriving.
Roam around in Reykjavik if you are not tired. Hotel Centrum is near the city center and most shops, cafes, restaurants are nearby and walking distance.
Northern Lights tour - Gateway to Iceland sounds great and they are a little expensive as compared to others but a mini bus is more personal experience and they also provide hot choclate which you will really appreciate once you are out in cold.
March 15th - I would recommend booking a Golden circle tour for this day (Pingvellir national park, Geysir hot spring & Gulfoss waterfall). It's almost a half day tour and the tour operators will pick you up and drop you back at the hotel. You can book online or just book using your hotel concierge or from teh tourism center near city center. This tour will be approximately ~40 - 50 pounds. DO TRY to eat at hotel geysir's restaurant near Geysir as their lamb chops, whale steak, lobster and beef carpaccio are amazing.
You can also check out city's nightlife as this will be a Saturday. Iceland is known for it's rock scene and Iceland waves (in oct) is one of their very popular music festivals. usually, some bars will have live music going on and you can check with the concierge. Otherwise you can check out Lebowski bar, Slipping bar at Marina, Hoftif night club, Austere club and several other bars in the same area in city center.
March 16th - You can book several tours this day or just decide to chill in Reykjavik, see the cathedral, the frozen lake, try rye bread ice cream at cafe Loki or eat at the restraurants near Marina or do some souvenir shopping.
If you do want to take another tour -
1) You can do the south shore tour which takes you to Skofgass waterfall and others, the beautfiful village of Vik and near the Eyjafjallajokull volcano that erupted in 2010 or Myrdalsjkull glacier. Some tours even take you to Hekla volcano. Different tour give you differrent options.
2) You can do a volcano and glacier (solheimajokull) walk in the north east of Reykjavik as well
3) You can also do a whale watching tour
March 17th - Use this day to either explore more of Reykjavik if you have not already or go to Blue Lagoon if you have not already or buy some lapopuesa (Icelandic sweater) or souveniers. Since your flight is at 4:30, you need to be at the airport by 2:30 /3 and hence need to catch the 2 PM bus from blue lagoon if you do it that day. In that case, you should plan on reaching at the earliest so you have a few hours at blue lagoon.
Carry a great camera and build up an appetite. Bon Voyage. Let me know if you need any more information or changes to your itinerary.