There is no way you can ever get to 1.120 with 10.5lbs of dry extract. According to Beersmith you should have a starting gravity of 1.085 with that.
Please note that hydrometers are calibrated at 60F (typically but not always) and any reading taken at any other temp needs to be adjusted (there is a tool for that in Beersmith).
Gravity readings should be taken when the cooled wort is in the fermentor. For extract brewing where you are boiling less than the full 5 gallons this is where you have the opportunity to adjust the O.G. by how much top up (pre-boiled or bottled) water you add. Generally you add water until you get the O.G. you want rather than a fixed volume.
I quickly made up a RIS recipe in Beersmith and it should look something like this (note the ton of hops required due to the super low utilization):
Recipe Specifications
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Boil Size: 3.67 gal
Post Boil Volume: 3.38 gal
Batch Size (fermenter): 5.00 gal
Bottling Volume: 4.60 gal
Estimated OG: 1.120 SG
Estimated Color: 44.2 SRM
Estimated IBU: 67.3 IBUs
Brewhouse Efficiency: 72.00 %
Est Mash Efficiency: 0.0 %
Boil Time: 60 Minutes
Ingredients:
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Amt Name Type # %/IBU
12.0 oz Roasted Barley (300.0 SRM) Grain 1 4.5 %
8.0 oz Black (Patent) Malt (500.0 SRM) Grain 2 3.0 %
8.0 oz Chocolate Malt (350.0 SRM) Grain 3 3.0 %
15 lbs Light Dry Extract (8.0 SRM) Dry Extract 4 89.6 %
4.00 oz Admiral [14.75 %] - Boil 60.0 min Hop 5 53.5 IBUs
2.00 oz Goldings, East Kent [6.47 %] - Boil 60.0 Hop 6 11.7 IBUs
1.00 oz Goldings, East Kent [6.47 %] - Boil 10.0 Hop 7 2.1 IBUs
2.0 pkg Nottingham (Danstar #-) [23.66 ml] Yeast 8 -
1.00 tsp Yeast Nutrient (Primary 3.0 days) Other 9 -
Mash Schedule: Single Infusion, Light Body, No Mash Out
Total Grain Weight: 16 lbs 12.0 oz
Also if you want to brew to this gravity you HAVE to have oxygen.
This is an advanced type of recipe and hard to pull off.