There are multiple threads on this site showing different approaches to take on it. Most sophisticated heat load tools use ACCA Manual-J, which is basically a refinement of older I=B=R methods, which is also a reasonable starting point.
Bu since you have a fuel-use history on this place, it's possible to put a stake in the ground for an upper bound using the old boiler as the measuring instrument. The boiler's nameplate input & output BTUs, a mid-winter fuel bill with exact billing dates, and a zip code (to look up weather data) is enough information to get started. I'll explain how that's done on a separate response if you can dig up that information.
The classic I=B=R method using the building assembly details works too. "Well insulated" does not have a definition that anybody can work from- wall type, siding type, and insulation type & thicknesses are needed to take as stab at it. Window type (and any storm windows), or the labeled U-factors if you have them are also critical. From the construction type one can estimate a U-factor for the wall area, window area, and roof/attic area. A U-factor is a linear approximation of the heat transfer per square foot of area of each of these elements per degree difference. eg: Your inside design temp is 70F, the 99% outdoor design temp for D.C. is +17F, so the temperature difference is 70-17= 53F.
Assuming the addition has 6 double hung windows that are 10 square feet each you have 60 square feet of window. If the manufacturer labeled it U0.43, the heat load of those windows is 60' x 53F x 0.43= 1367 BTU/hr.
Say the addition was 2x6 framed with R19 batts, wood siding. R=1/U, thus U=1/R, but the thermal bridging of the studs reduces the average R to something like R13 after thermal bridging of all the studs/plates/headers etc., (I can give multiple sources on that, if you like- best you're going to do is R14 average unless you use advanced framing techniques.) That yield a U-factor for the walls of 1/R13= 0.077. Say you have 160' of running wall 10' high for 1600 square feet of wall, less 60' of windows, is 1540' of U0.077 wall. The heat load of the walls are then 1540' x 53F x 0.077= 6285 BTU/hr.
Then, say it's a truss roof with 12" of blown cellulose. The cellulose itself is about R42, but cut that by 10% or so for the thermal bridging of the truss elements, call it R38 (which would be current code min), or a U-factor of 1/38= 0.026 If the attic area is 1000 square feet, the heat load of the attic is 1000' x 53F x 0.026 = 1378 BTU/hr
etc etc etc
Then throw in a WAG for the natural infiltration rates, call it 10 cubic feet per minute. That's 600 cubic feet per hour. The specific heat of a cubic foot of air is about 0.18 BTU per degree F, so the heat load from infiltration is 600 x 53F x 0.18= 5724 BTU/hr
Add it all up. That's something of an upper bound for the true heat load.
Then start subtracting for hot bodies & 24/7 plug loads. Every sleeping human is worth 250BTU/hr, a refrigerator is another 200 BTU/hr. A Tivo is about 300BTU/hr. it goes on.
Brick walls perform differently base on the actual wall construction, so let's have it- what type of brick or block, solid vs. hollow core, how big, and is there a vent cavity between inner & outer wythes, etc. Most block walls are good for at least R1.5 (U0.67) some are good for R3 (U-0.33) and it'll make a real difference in the final number.
Poured concrete basement walls are about U1, but only count the area down to about a foot below grade, ignore the rest.
Single pane windows estimate at U1 With tight fitting storm windows that drops to U0.5. Cheap 1970s-80s vintage double panes are maybe U0.6 or 0.7. Better 1980s double-panes with a low-E coating estimate at U0.40
Typical 2x4 fiberglass insulated walls estimate at U0.01 for rough sizing. (With siding type and actual insulation-R that can be refined a bit.) Typical 2x6 walls figure U0.077.
Solid core 2" doors run about U0.5, six panel wood doors are around U0.8. (Insulated steel and fiber glass doors you can look up by model- they vary quite a bit.)
Build yourself spreadsheet on a room by room heat loss listing all exterior wall/ceiling components components seperately, and sum them up both by room, and the whole house.
Somewhere near the bottom line you'll discover that as long as there's glass in the windows and doors that close there's NO WAY you'd ever need a 300KBTU/hr boiler, or even a 100K boiler in a house that size in that climate.
If you're buying a gas-fire modulating condensing boiler heating hot water with an indirect-fired tank off the boiler is absolutely the right way to go. If you're going with decent efficiency gas-fired cast iron boiler that makes sense too. If oil or propane are your only fuel options, you'll be better off with a heat-pump electric water heater (which will also help dehumidify the basement) or even a plain-old 0.90EF electric tank. There's no way that an embedded coil like you currently have will ever make sense, especially with a boiler correctly sized for the space heating load. The off-season efficiency is low, and the capacity will also be low with a much smaller boiler. And yes, you WILL have a much smaller boiler.
Without knowing much more about your house than you've given so far I'd hazard you're under 80KBTU/hr and with reasonable insulation and air sealing upgrades (and low-E storms over any remaining single-panes) you'd come in under 60KBTU/hr, maybe even under 50KBTU/hr.
Once you have a handle on the heat load, tell us how much radiation (and type) there is, by zone (it matters, from a system design point of view.)