obtenha o documento com $box com mongodb e adicione o campo de distância para cada registro com ponto de coordenada específico

Este deve funcionar:

db.collection.aggregate([
   {
      "$match": {
         "location": {
            "$geoWithin": {
               "$box": [
                  [0.024719919885622943, 51.54643953472475],
                  [-0.1589577534542408, 51.47239969138267]
               ]
            }
         }
      }
   },
   { $set: { lon: -0.0649729793707321 } },
   { $set: { lat: 51.50160291888072 } },
   {
      $set: {
         distance: {
            $let: {
               vars: {
                  dlon: { $degreesToRadians: { $subtract: [{ $arrayElemAt: ["$location.coordinates", 0] }, "$lon"] } },
                  dlat: { $degreesToRadians: { $subtract: [{ $arrayElemAt: ["$location.coordinates", 1] }, "$lat"] } },
                  lat1: { $degreesToRadians: { $arrayElemAt: ["$location.coordinates", 1] } },
                  lat2: { $degreesToRadians: "$lat" }
               },
               in: {
                  // Haversine formula: sin²(dLat / 2) + sin²(dLon / 2) * cos(lat1) * cos(lat2);
                  $add: [
                     { $pow: [{ $sin: { $divide: ["$$dlat", 2] } }, 2] },
                     { $multiply: [{ $pow: [{ $sin: { $divide: ["$$dlon", 2] } }, 2] }, { $cos: "$$lat1" }, { $cos: "$$lat2" }] }
                  ]
               }
            }
         }
      }
   },
   {
      $set: {
         distance: {
            // Distance in Meters given by "6372.8 * 1000"
            $multiply: [6372.8, 1000, 2, { $asin: { $sqrt: "$distance" } }]
         }
      }
   },
])

Apenas uma observação, $box é usado apenas para Pares de coordenadas herdados . Para consulta adequada (ou seja, uso adequado de 2dsphere index) você deve usar $geometry :

db.collection.createIndex({ location: "2dsphere" })
db.collection.find(
   {
      "location": {
         "$geoWithin": {
            "$geometry": {
               type: "Polygon",
               coordinates: [[
                  [-0.1589577534542408, 51.47239969138267],
                  [-0.1589577534542408, 51.54643953472475],
                  [0.024719919885622943, 51.54643953472475],
                  [0.024719919885622943, 51.47239969138267],
                  [-0.1589577534542408, 51.47239969138267]
               ]]
            }
         }
      }
   }
).explain().queryPlanner.winningPlan.inputStage.indexName

--> location_2dsphere // which means index is used


db.collection.find(
   {
      "location": {
         "$geoWithin": {
            "$box": [
               [0.024719919885622943, 51.54643953472475],
               [-0.1589577534542408, 51.47239969138267]
            ]
         }
      }
   }
).explain().queryPlanner.winningPlan.stage

--> COLLSCAN // which means full collection scan

Ou uso adequado das Coordenadas Legadas:

db.collection.createIndex({ "location.coordinates": "2d" })
db.collection.find(
   {
      "location.coordinates": { // <- note this difference
         "$geoWithin": {
            "$box": [
               [0.024719919885622943, 51.54643953472475],
               [-0.1589577534542408, 51.47239969138267]
            ]
         }
      }
   }
).explain().queryPlanner.winningPlan.inputStage.indexName

--> location.coordinates_2d